Probability

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Summary

Summary of Probability Chapter

Theoretical Probability:
- Defined as:
  
  P(E) = Number of outcomes favourable to E / Number of all possible outcomes
- Assumes outcomes are equally likely.
Probability Values:
- Probability of a sure event = 1
- Probability of an impossible event = 0
- For any event E, 0 ≤ P(E) ≤ 1
Elementary Events:
- An event with only one outcome is called an elementary event.
- The sum of probabilities of all elementary events = 1.
Complementary Events:
- For any event E, P(E) + P(not E) = 1.
Experimental vs Theoretical Probability:
- Experimental probability is based on actual outcomes, while theoretical probability is based on assumptions.
Equally Likely Outcomes:
- Outcomes are not always equally likely (e.g., drawing from a bag with different colored balls).
Examples:
- Coin toss: P(head) = 1/2, P(tail) = 1/2.
- Drawing a ball from a bag with 3 red and 5 black balls: P(red) = 3/8, P(not red) = 5/8.
Important Notes:
- Theoretical probability was defined by Pierre Simon Laplace in 1795.
- Probability theory has applications in various fields including biology, economics, and physics.

Learning Objectives

Understand the definition of theoretical (classical) probability.
Calculate the probability of an event based on favorable outcomes and total outcomes.
Differentiate between sure events, impossible events, and elementary events.
Recognize the relationship between complementary events and their probabilities.
Apply the concept of equally likely outcomes in various probability experiments.
Analyze and solve problems involving empirical and theoretical probabilities.
Identify common misconceptions related to probability calculations.

Detailed Notes

Probability Notes

Key Concepts

Theoretical Probability:
- Defined as:
  
  P(E) = $\frac{\text{Number of outcomes favourable to E}}{\text{Number of all possible outcomes}}$
- Assumes outcomes are equally likely.
Types of Events:
- Sure Event: Probability = 1
- Impossible Event: Probability = 0
- Elementary Event: An event with only one outcome.
- Complementary Events: For any event E, $P(E) + P(not E) = 1$
Probability Range:
- $0 \leq P(E) \leq 1$

Examples

Coin Toss:
- Outcomes: Head (H) or Tail (T)
- Probability of Head: $P(H) = \frac{1}{2}$
- Probability of Tail: $P(T) = \frac{1}{2}$
Die Roll:
- Outcomes: 1, 2, 3, 4, 5, 6
- Probability of rolling a 3: $P(3) = \frac{1}{6}$

Important Formulas

Probability of an Event:
$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total outcomes}}$
Complementary Probability:
$P(E) = 1 - P(not E)$

Common Mistakes

Misunderstanding equally likely outcomes. Not all experiments have equally likely outcomes (e.g., drawing from a bag with different colored balls).
Assuming probabilities can be negative or exceed 1.

Tips

Always check if outcomes are equally likely before applying theoretical probability.
Remember that the sum of probabilities of all possible outcomes must equal 1.

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips in Probability

Common Pitfalls

Misunderstanding Equally Likely Outcomes: Students often assume that all outcomes in an experiment are equally likely without verifying. For example, when drawing a ball from a bag containing different colored balls, the outcomes are not equally likely if the number of each color is different.
Incorrect Probability Calculation: Some students calculate probabilities based on the number of outcomes rather than the number of favorable outcomes. For instance, when calculating the probability of rolling a specific number on a die, they might mistakenly think there are more outcomes than there actually are.
Confusing Complementary Events: Students may forget that the sum of the probabilities of an event and its complement equals 1. For example, if the probability of an event E is 0.3, the probability of not E should be calculated as 1 - 0.3 = 0.7.
Assuming Independence Incorrectly: In problems involving multiple events, students sometimes incorrectly assume that events are independent when they are not. For example, drawing cards from a deck without replacement affects the probabilities of subsequent draws.

Tips for Success

Always Define Your Sample Space: Clearly outline all possible outcomes before calculating probabilities. This helps in identifying favorable outcomes accurately.
Check for Equally Likely Outcomes: Before applying the formula for probability, ensure that the outcomes are indeed equally likely. If not, adjust your calculations accordingly.
Use Complementary Events: If calculating the probability of an event seems complex, consider using the complementary event to simplify your calculations.
Practice with Real-Life Examples: Engage with practical examples, such as games of chance or everyday decisions, to better understand probability concepts.
Review Common Probability Formulas: Familiarize yourself with key probability formulas and definitions, such as the probability of an event being the number of favorable outcomes divided by the total number of outcomes.

Practice & Assessment

Multiple Choice Questions

\frac{3}{13}

\frac{1}{4}

\frac{1}{13}

\frac{9}{52}

Correct Answer: A

Solution:

There are 12 face cards in a deck (3 face cards per suit and 4 suits). Therefore, the probability is

\frac{12}{52} = \frac{3}{13}

\frac{1}{8}

\frac{1}{4}

\frac{1}{2}

\frac{3}{8}

Correct Answer: A

Solution:

The probability of getting heads on a single toss is \frac{1}{2}. Therefore, the probability of getting heads on all three tosses is \left(\frac{1}{2}\right)^3 = \frac{1}{8}.

\frac{1}{5}

\frac{1}{2}

\frac{1}{4}

\frac{3}{10}

Correct Answer: B

Solution:

The probability of drawing a green ball (neither red nor blue) is \frac{3}{5 + 3 + 2} = \frac{3}{10}.

\frac{1}{4}

\frac{1}{2}

\frac{1}{8}

\frac{1}{3}

Correct Answer: A

Solution:

The numbers greater than 6 on the spinner are 7 and 8. There are 2 favorable outcomes. The total number of possible outcomes is 8. Therefore, the probability is \frac{2}{8} = \frac{1}{4}.

1/6

1/3

1/2

1/5

Correct Answer: A

Solution:

The probability of rolling a 'D' is the number of 'D' faces divided by the total number of faces. Therefore,

P(D) = \frac{1}{6}

\frac{3}{8}

\frac{5}{8}

\frac{1}{8}

\frac{1}{2}

Correct Answer: A

Solution:

The total number of balls is 3 + 5 = 8. The number of red balls is 3. Therefore, the probability of drawing a red ball is \frac{3}{8}.

0.25

0.5

0.75

1.0

Correct Answer: B

Solution:

The numbers less than 5 are 1, 2, 3, and 4. So, the probability is

\frac{4}{8} = 0.5

\frac{1}{6}

\frac{1}{3}

\frac{1}{2}

\frac{2}{3}

Correct Answer: B

Solution:

The numbers greater than 4 on a die are 5 and 6. There are 2 favorable outcomes out of 6 possible outcomes. Thus, P(number > 4) = \frac{2}{6} = \frac{1}{3}.

\frac{1}{8}

\frac{3}{8}

\frac{1}{2}

\frac{5}{8}

Correct Answer: B

Solution:

The possible outcomes when a coin is tossed three times are HHH, HHT, HTH, THH, TTH, THT, HTT, TTT. The number of outcomes with exactly two heads is 3 (HHT, HTH, THH). Therefore, the probability is

\frac{3}{8}

0.25

0.5

0.75

Correct Answer: C

Solution:

The possible outcomes when a coin is tossed twice are HH, HT, TH, TT. The event of getting at least one tail includes HT, TH, and TT. Thus,

P(\text{at least one tail}) = \frac{3}{4} = 0.75

\frac{1}{4}

\frac{1}{2}

\frac{3}{4}

\frac{1}{3}

Correct Answer: A

Solution:

The possible outcomes when a coin is tossed twice are HH, HT, TH, TT. Only one outcome is two heads (HH), so the probability is

\frac{1}{4}

\frac{2}{5}

\frac{3}{10}

\frac{1}{2}

\frac{1}{5}

Correct Answer: B

Solution:

The probability of drawing the first red ball is

\frac{4}{5}

. After drawing one red ball, there are 3 red balls left out of a total of 4 balls. Hence, the probability of drawing another red ball is

\frac{3}{4}

. Therefore, the probability of both balls being red is

\frac{4}{5} \times \frac{3}{4} = \frac{3}{5} \times \frac{1}{2} = \frac{3}{10}

0.2

0.8

0.5

0.1

Correct Answer: A

Solution:

The probability of drawing a blue ball, which is not red, is the number of blue balls divided by the total number of balls. Thus, P(not red) = 1/5 = 0.2.

0.231

0.25

0.192

0.308

Correct Answer: A

Solution:

There are 12 face cards in a deck of 52 cards. Thus, the probability is

P(\text{face card}) = \frac{12}{52} = \frac{3}{13} \approx 0.231

1/8

3/8

1/2

1/4

Correct Answer: B

Solution:

The possible outcomes are HHT, HTH, THH. There are 3 favorable outcomes. Total possible outcomes are 8. So, the probability is

\frac{3}{8}

0.1389

0.8611

0.5556

0.4444

Correct Answer: B

Solution:

The probability of drawing a non-defective pen is the ratio of non-defective pens to the total number of pens. Thus,

P(\text{not defective}) = \frac{144 - 20}{144} = \frac{124}{144} = 0.8611

1/5

4/5

1/4

1/2

Correct Answer: A

Solution:

There are 5 balls in total, and only 1 of them is blue. Therefore, the probability of drawing a blue ball is 1/5.

1/3

3/9

3/10

1/9

Correct Answer: C

Solution:

The total number of balls is 4 + 3 + 2 = 9. The number of blue balls is 3. Therefore, the probability of drawing a blue ball is 3/9 = 1/3.

\frac{1}{4}

\frac{1}{2}

\frac{1}{3}

\frac{1}{13}

Correct Answer: A

Solution:

There are 13 spades in a deck of 52 cards. Therefore, the probability of drawing a spade is

\frac{13}{52} = \frac{1}{4}

0.25

0.5

0.75

Correct Answer: B

Solution:

There are two possible outcomes when a coin is tossed: head or tail. The probability of getting a tail is

P(\text{tail}) = \frac{1}{2} = 0.5

1/2

3/8

5/8

1/4

Correct Answer: B

Solution:

The prime numbers between 1 and 8 are 2, 3, 5, and 7. Thus, there are 4 prime numbers. The probability is 4/8 = 1/2.

1/8

1/4

1/2

3/8

Correct Answer: A

Solution:

The possible outcomes for each toss are H or T. Therefore, for three tosses, the total number of outcomes is 2^3 = 8. Only one outcome is HHH. Thus, the probability is 1/8.

1/4

1/2

1/8

1/3

Correct Answer: A

Solution:

The numbers greater than 6 are 7 and 8. Therefore, the probability is

\frac{2}{8} = \frac{1}{4}

\frac{3}{5}

\frac{1}{2}

\frac{2}{5}

\frac{1}{3}

Correct Answer: C

Solution:

There are 30 students, with 12 being girls. The probability of choosing a girl is

\frac{12}{30} = \frac{2}{5}

1/3

1/2

2/3

5/6

Correct Answer: C

Solution:

The numbers greater than 2 on a die are 3, 4, 5, and 6. Thus, there are 4 favorable outcomes. The probability is

\frac{4}{6} = \frac{2}{3}

4/17

5/17

8/17

9/17

Correct Answer: C

Solution:

The probability of drawing a white marble is given by the ratio of the number of white marbles to the total number of marbles, which is

\frac{8}{17}

\frac{99}{100}

\frac{1}{100}

\frac{1}{10}

\frac{1}{2}

Correct Answer: A

Solution:

The probability of not winning is the complement of the probability of winning. Thus, it is

1 - \frac{1}{100} = \frac{99}{100}

1/6

1/36

5/36

1/12

Correct Answer: A

Solution:

The possible outcomes when two dice are thrown are 36. The combinations that result in a sum of 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), which are 6 in total. Therefore, the probability is 6/36 = 1/6.

\frac{9}{17}

\frac{8}{17}

\frac{5}{17}

\frac{12}{17}

Correct Answer: A

Solution:

The total number of marbles is 17. The number of non-white marbles is 5 + 4 = 9. Thus, P(not white) = \frac{9}{17}.

\frac{1}{4}

\frac{1}{8}

\frac{1}{2}

\frac{1}{3}

Correct Answer: A

Solution:

There are 2 outcomes for each toss: head (H) or tail (T). The possible outcomes for 3 tosses are HHH, HHT, HTH, THH, TTH, THT, HTT, TTT. Only HHH and TTT result in the same outcome. Thus, the probability is

\frac{2}{8} = \frac{1}{4}

0.33

0.5

0.67

0.17

Correct Answer: A

Solution:

The numbers less than 3 on a die are 1 and 2. So, the probability is

P(\text{less than 3}) = \frac{2}{6} = \frac{1}{3} = 0.33

1/2

1/4

3/8

5/8

Correct Answer: A

Solution:

The even numbers on the spinner are 2, 4, 6, and 8, which are 4 in total. Therefore, the probability is 4/8 = 1/2.

0.38

0.48

0.58

0.42

Correct Answer: A

Solution:

Since the events are complementary, P(Reshma wins) = 1 - P(Sangeeta wins) = 1 - 0.62 = 0.38.

1/3

1/2

1/6

2/3

Correct Answer: A

Solution:

The numbers greater than 4 on a die are 5 and 6. Therefore, there are 2 favorable outcomes. The probability is 2/6 = 1/3.

\frac{3}{8}

\frac{5}{8}

\frac{1}{2}

\frac{1}{4}

Correct Answer: A

Solution:

The probability of drawing a red ball (not black) is the number of red balls divided by the total number of balls. Thus, the probability is

\frac{3}{3+5} = \frac{3}{8}

1/6

1/2

1/3

2/3

Correct Answer: B

Solution:

The possible outcomes when a die is thrown are 1, 2, 3, 4, 5, and 6. The numbers less than 4 are 1, 2, and 3. Therefore, the probability is

\frac{3}{6} = \frac{1}{2}

3/13

1/4

1/13

9/52

Correct Answer: A

Solution:

There are 12 face cards in a deck of 52 cards (4 Jacks, 4 Queens, 4 Kings). Therefore, the probability is 12/52 = 3/13.

\frac{1}{9}

\frac{1}{12}

\frac{1}{8}

\frac{1}{18}

Correct Answer: B

Solution:

The favorable outcomes for a sum of 9 are (3,6), (4,5), (5,4), and (6,3). Thus, there are 4 favorable outcomes. The total number of outcomes when a die is thrown twice is 36. Therefore, the probability is

\frac{4}{36} = \frac{1}{9}

1/3

1/2

1/6

1/4

Correct Answer: B

Solution:

The die has 6 faces, and 'A' appears on 2 of them. Therefore, the probability is

\frac{2}{6} = \frac{1}{3}

1/4

1/2

3/4

1/3

Correct Answer: C

Solution:

The possible outcomes are HH, HT, TH, TT. The outcomes with at least one head are HH, HT, TH, which are 3 in total. Therefore, the probability is 3/4.

\frac{1}{3}

\frac{1}{2}

\frac{1}{6}

\frac{2}{3}

Correct Answer: A

Solution:

The die has 6 faces, with the letter 'A' appearing on 2 of them. Therefore, the probability of getting 'A' is

\frac{2}{6} = \frac{1}{3}

1/4

1/3

1/2

3/4

Correct Answer: C

Solution:

The possible outcomes when two coins are tossed are HH, HT, TH, TT. The outcomes with exactly one head are HT and TH. Therefore, the probability is 2/4 = 1/2.

1/6

1/3

1/2

2/3

Correct Answer: B

Solution:

The numbers less than 3 on a die are 1 and 2. Thus, there are 2 favorable outcomes. The probability is

\frac{2}{6} = \frac{1}{3}

\frac{1}{3}

\frac{1}{2}

\frac{1}{6}

\frac{2}{3}

Correct Answer: B

Solution:

The die has 6 faces with letters A, B, C, D, E, and A. The probability of getting 'A' is the number of 'A' faces divided by the total number of faces. Since there are 2 'A' faces, the probability is

\frac{2}{6} = \frac{1}{3}

\frac{1}{8}

\frac{1}{4}

\frac{3}{8}

\frac{1}{2}

Correct Answer: B

Solution:

The multiples of 4 in the range 1 to 8 are 4 and 8. Therefore, there are 2 favorable outcomes. The probability is \frac{2}{8} = \frac{1}{4}.

1/4

1/2

1/8

3/8

Correct Answer: A

Solution:

The numbers greater than 6 are 7 and 8. Therefore, there are 2 favorable outcomes. The probability is 2/8 = 1/4.

\frac{5}{6}

\frac{7}{8}

\frac{31}{36}

\frac{17}{18}

Correct Answer: A

Solution:

The number of good pens is 144 - 20 = 124. The probability of drawing a good pen is

\frac{124}{144} = \frac{31}{36}

1/2

1/3

1/6

2/3

Correct Answer: A

Solution:

The possible outcomes when a die is thrown are 1, 2, 3, 4, 5, 6. The even numbers are 2, 4, and 6. Therefore, the probability of getting an even number is the number of favorable outcomes (3) divided by the total number of outcomes (6), which is 1/2.

\frac{1}{9}

\frac{1}{12}

\frac{1}{8}

\frac{1}{6}

Correct Answer: B

Solution:

To find the probability of the sum being 9, we list all possible outcomes: (3,6), (4,5), (5,4), (6,3). There are 4 favorable outcomes. Total possible outcomes when a die is thrown twice is 6 \times 6 = 36. Thus, the probability is \frac{4}{36} = \frac{1}{9}.

\frac{1}{3}

\frac{1}{2}

\frac{2}{5}

\frac{3}{10}

Correct Answer: C

Solution:

The multiples of 3 between 1 and 10 are 3, 6, and 9. There are 3 favorable outcomes. Total outcomes are 10. Thus, the probability is \frac{3}{10}.

\frac{1}{8}

\frac{1}{2}

\frac{3}{8}

\frac{5}{8}

Correct Answer: B

Solution:

The odd numbers on the spinner are 1, 3, 5, and 7. There are 4 odd numbers out of 8 possible outcomes, so the probability is

\frac{4}{8} = \frac{1}{2}

0.25

0.5

0.75

1.0

Correct Answer: C

Solution:

The possible outcomes are HH, HT, TH, TT. The probability of getting at least one head is

\frac{3}{4} = 0.75

\frac{1}{7}

\frac{1}{3}

\frac{1}{5}

\frac{1}{2}

Correct Answer: A

Solution:

The probability of drawing two blue marbles is \frac{\binom{6}{2}}{\binom{15}{2}} = \frac{15}{105} = \frac{1}{7}.

0.1667

0.3333

0.5

0.6667

Correct Answer: B

Solution:

The numbers greater than 4 on a die are 5 and 6. Thus, the probability is

P(\text{greater than 4}) = \frac{2}{6} = \frac{1}{3} = 0.3333

0.25

0.5

0.75

Correct Answer: B

Solution:

The probability of getting a tail when a coin is tossed is

P(\text{tail}) = \frac{1}{2} = 0.5

0.375

0.5

0.625

0.25

Correct Answer: A

Solution:

The probability of drawing a red ball is given by the ratio of the number of red balls to the total number of balls. Thus,

P(\text{red}) = \frac{3}{3 + 5} = \frac{3}{8} = 0.375

True or False

Correct Answer: False

Solution:

When two coins are tossed, the possible outcomes are HH, HT, TH, and TT. The probability of getting two heads is

\frac{1}{4}

, not

\frac{1}{3}

Correct Answer: False

Solution:

There are 4 red balls and 1 blue ball, making the probability of drawing a red ball

\frac{4}{5}

and a blue ball

\frac{1}{5}

. These outcomes are not equally likely.

Correct Answer: False

Solution:

When two coins are tossed, there are four possible outcomes: HH, TT, HT, and TH. Therefore, the statement is incorrect.

Correct Answer: True

Solution:

There are 4 red balls and 1 blue ball, making the probability of drawing a red ball 4/5, which is greater than the probability of drawing a blue ball, 1/5.

Correct Answer: True

Solution:

A fair die means that each face has an equal chance of landing face up. Therefore, each number from 1 to 6 has a probability of

\frac{1}{6}

Correct Answer: False

Solution:

When two coins are tossed, the possible outcomes are HH, TT, and HT (or TH). The probability for each outcome is actually

\frac{1}{4}

for HH,

\frac{1}{4}

for TT, and

\frac{1}{2}

for HT or TH, since HT and TH are distinct outcomes.

Correct Answer: True

Solution:

A sure event is certain to happen, so its probability is 1.

Correct Answer: True

Solution:

The probability of an event E, denoted as P(E), is defined such that 0 \leq P(E) \leq 1. This means that the probability of any event is always a number between 0 and 1, inclusive.

Correct Answer: True

Solution:

A fair die has three odd numbers (1, 3, 5) and three even numbers (2, 4, 6). Thus, the probability of rolling an odd number is 3/6 = 0.5.

Correct Answer: False

Solution:

In a fair coin toss, the probability of getting a head is equal to the probability of getting a tail, both being

\frac{1}{2}

Correct Answer: True

Solution:

According to the definition of probability, it is a number between 0 and 1, inclusive.

Correct Answer: True

Solution:

The probability of an event is defined as a number

P(E)

such that

0 \leq P(E) \leq 1

Correct Answer: True

Solution:

Probability values range from 0 (impossible event) to 1 (certain event).

Correct Answer: True

Solution:

A fair die has numbers 1 to 6, all of which are less than 7, so the probability is 1.

Correct Answer: False

Solution:

The probability of an impossible event is 0, not 1. A probability of 1 indicates a certain event.

Correct Answer: True

Solution:

The probability of drawing a red ball is

\frac{4}{5} = 0.8

, as there are 4 red balls out of a total of 5 balls.

Correct Answer: False

Solution:

When two coins are tossed, the possible outcomes are HH, TT, HT, and TH. Therefore, there are four possible outcomes, not three.

Correct Answer: True

Solution:

A sure event is one that is certain to happen, and its probability is always 1.

Correct Answer: True

Solution:

An event that is certain to happen has a probability of 1.

Correct Answer: True

Solution:

A fair die has six faces with numbers 1 to 6. The even numbers are 2, 4, and 6. Thus, the probability of rolling an even number is

\frac{3}{6} = 0.5

Correct Answer: False

Solution:

The probability of an impossible event is 0, not 1.

Correct Answer: True

Solution:

If a bag contains only red balls, then drawing a ball will always result in a red ball, making the probability 1.

Correct Answer: True

Solution:

A fair coin has two equally likely outcomes: heads or tails. Therefore, the probability of getting heads is

\frac{1}{2} = 0.5

Correct Answer: True

Solution:

The probability of not getting a 5 in one throw is \frac{5}{6}. Thus, the probability of not getting a 5 in two throws is \left(\frac{5}{6}\right)^2 = \frac{25}{36}. Therefore, the probability of getting at least one 5 is 1 - \frac{25}{36} = \frac{11}{36}, which is greater than 0.5.

Correct Answer: True

Solution:

A fair die has six faces numbered from 1 to 6, and each face has an equal chance of landing face up. Therefore, the probability of each number appearing is equal.

Correct Answer: True

Solution:

A standard die has numbers 1 to 6, so there is no possibility of getting a number greater than 6. Therefore, the probability is 0.

Correct Answer: True

Solution:

For any event

E

, the sum of the probabilities of the event and its complement is 1, i.e.,

P(E) + P(\text{not } E) = 1

Correct Answer: True

Solution:

A fair coin has two equally likely outcomes: heads or tails, each with a probability of 0.5.

Correct Answer: True

Solution:

A die has 6 faces numbered 1 to 6. The odd numbers are 1, 3, and 5, so there are 3 favorable outcomes. The probability is

\frac{3}{6} = \frac{1}{2}

Correct Answer: True

Solution:

The sum of the probabilities of all the elementary events of an experiment is 1, as they represent all possible outcomes.

Correct Answer: True

Solution:

The die has 6 faces, with 'A' appearing on 2 of them. Therefore, the probability of rolling an 'A' is 2/6, which simplifies to 1/3.

Correct Answer: True

Solution:

When a fair coin is tossed, there are two equally likely outcomes: heads or tails. Thus, the probability of getting a head is

\frac{1}{2} = 0.5

Correct Answer: False

Solution:

The probability of an impossible event is 0, as it cannot occur.

Correct Answer: True

Solution:

A die has 6 faces with numbers 1, 2, 3, 4, 5, and 6. The odd numbers are 1, 3, and 5. Therefore, the probability of getting an odd number is

\frac{3}{6} = \frac{1}{2}

Correct Answer: True

Solution:

The probability of getting a sum of 7 is 6/36 (as there are 6 combinations), whereas the probability of getting a sum of 2 is 1/36 (only one combination). Therefore, getting a sum of 7 is more likely.

Correct Answer: True

Solution:

When a coin is tossed three times, the total number of outcomes is

2^3 = 8

. The outcome of getting three heads (HHH) is just one of these outcomes, so the probability is

\frac{1}{8}

Correct Answer: True

Solution:

By definition, the sum of the probabilities of all elementary events in an experiment equals 1.

Correct Answer: True

Solution:

A standard die has numbers 1 through 6, so it is impossible to roll a number greater than 6. Thus, the probability is 0.

Correct Answer: True

Solution:

When a fair coin is tossed, there are two possible outcomes: head or tail. Since these are the only outcomes, the probability of getting either a head or a tail is 1.

Probability

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Summary

Summary of Probability Chapter

Learning Objectives

Learning Objectives

Detailed Notes

Probability Notes

Key Concepts

Examples

Important Formulas

Common Mistakes

Tips

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips in Probability

Common Pitfalls

Tips for Success

Practice & Assessment

Multiple Choice Questions

Q1.A deck of cards is shuffled and one card is drawn at random. What is the probability that it is a face card (Jack, Queen, or King)?

Correct Answer: A

Q2.A game consists of tossing a coin three times. What is the probability that all three tosses result in heads?

Correct Answer: A

Q3.A bag contains 5 red balls, 3 green balls, and 2 blue balls. If a ball is drawn at random, what is the probability that it is neither red nor blue?

Correct Answer: B

Q4.In a game, a spinner is divided into 8 equal sections numbered 1 to 8. What is the probability that the spinner lands on a number greater than 6?

Correct Answer: A

Q5.A child has a die with faces labeled A, B, C, D, E, and A. What is the probability of rolling a 'D'?

Correct Answer: A

Q6.A box contains 3 red balls and 5 black balls. A ball is drawn at random. What is the probability that the ball drawn is red?

Correct Answer: A

Q7.In a game, a spinner is divided into 8 equal sections numbered 1 to 8. What is the probability that it will point at a number less than 5?

Correct Answer: B

Q8.A die is thrown once. What is the probability of getting a number greater than 4?

Correct Answer: B

Q9.A game consists of tossing a fair coin three times. What is the probability of getting exactly two heads?

Correct Answer: B

Q10.In a game, a coin is tossed twice. What is the probability of getting at least one tail?

Correct Answer: C

Q11.A game consists of tossing a coin twice. What is the probability of getting two heads?

Correct Answer: A

Q12.A bag contains 4 red balls and 1 blue ball. If two balls are drawn at random without replacement, what is the probability that both balls are red?

Correct Answer: B

Q13.A bag contains 4 red balls and 1 blue ball. If a ball is drawn at random, what is the probability that it is not a red ball?

Correct Answer: A

Q14.In a standard deck of 52 cards, what is the probability of drawing a face card?

Correct Answer: A

Q15.A game consists of tossing a one rupee coin 3 times. What is the probability of getting exactly two heads?

Correct Answer: B

Q16.A lot consists of 144 ball pens of which 20 are defective. What is the probability that a pen drawn at random is not defective?

Correct Answer: B

Q17.A bag contains 4 red balls and 1 blue ball. What is the probability of drawing a blue ball?

Correct Answer: A

Q18.If a bag contains 4 red balls, 3 blue balls, and 2 green balls, what is the probability of drawing a blue ball?

Correct Answer: C

Q19.In a deck of 52 cards, what is the probability of drawing a spade?

Correct Answer: A

Q20.A coin is tossed once. What is the probability of getting a tail?

Correct Answer: B

Q21.In a game, a spinner is divided into 8 equal sections numbered 1 to 8. What is the probability that the spinner lands on a prime number?

Correct Answer: B

Q22.In a game, a coin is tossed three times. What is the probability that all tosses result in heads?

Correct Answer: A

Q23.A game consists of spinning a wheel with 8 equal sections numbered 1 to 8. What is the probability that it will land on a number greater than 6?

Correct Answer: A

Q24.In a class of 30 students, each student is equally likely to be chosen as the class representative. If 18 students are boys and the rest are girls, what is the probability that a girl is chosen as the representative?

Correct Answer: C

Q25.A die is thrown once. What is the probability of getting a number greater than 2?

Correct Answer: C

Q26.A box contains 5 red marbles, 8 white marbles, and 4 green marbles. What is the probability of drawing a white marble?

Correct Answer: C

Q27.In a lottery, there are 100 tickets and only one of them is a winning ticket. If you buy one ticket, what is the probability that it is not a winning ticket?

Correct Answer: A

Q28.A fair die is thrown twice. What is the probability that the sum of the numbers on the two dice will be 7?

Correct Answer: A

Q29.A box contains 5 red marbles, 8 white marbles, and 4 green marbles. If one marble is drawn at random, what is the probability that it is not white?

Correct Answer: A

Q30.If a fair coin is tossed 3 times, what is the probability that all tosses result in the same outcome (either all heads or all tails)?

Correct Answer: A

Q31.A die is thrown once. What is the probability of getting a number less than 3?