35 Unit 11 Probability And Statistics Homework 2 Theoretical Probability Answers

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Unit 11 Probability and Statistics Homework 2 Theoretical Probability Answers

Introduction

Probability and statistics are essential branches of mathematics that help us understand and make predictions about the world around us. In this article, we will provide the answers to Homework 2 of Unit 11, focusing on theoretical probability. By delving into the questions and their solutions, we aim to enhance your understanding of this topic and prepare you for future statistical challenges.

Question 1: Rolling a Fair Die

In this question, we are asked to find the probability of rolling a number less than 4 on a fair six-sided die.

Solution:

The die has six faces, numbered 1 to 6. Out of these, the numbers less than 4 are 1, 2, and 3. Therefore, the favorable outcomes are 3. Since there are a total of 6 equally likely outcomes, the probability of rolling a number less than 4 is 3/6, which simplifies to 1/2 or 50%.

Question 2: Drawing a Card

This question involves drawing a card from a standard deck of 52 playing cards and finding the probability of drawing a heart.

Solution:

A standard deck of playing cards contains 52 cards, out of which 13 are hearts. Therefore, the favorable outcomes are 13. Since there are 52 equally likely outcomes, the probability of drawing a heart is 13/52, which simplifies to 1/4 or 25%.

Question 3: Flipping Coins

In this question, we are asked to find the probability of flipping two coins and getting at least one head.

Solution:

When flipping two coins, there are four possible outcomes: both heads (HH), both tails (TT), a head and a tail (HT), or a tail and a head (TH). Out of these four outcomes, three have at least one head (HH, HT, TH). Therefore, the favorable outcomes are 3. Since there are a total of 4 equally likely outcomes, the probability of getting at least one head when flipping two coins is 3/4 or 75%.

Question 4: Drawing Marbles

This question involves drawing marbles from a bag containing 5 red marbles, 3 blue marbles, and 2 green marbles. We need to find the probability of drawing a blue marble.

Solution:

The bag contains a total of 5 + 3 + 2 = 10 marbles. Out of these, there are 3 blue marbles. Therefore, the favorable outcomes are 3. Since there are 10 equally likely outcomes, the probability of drawing a blue marble is 3/10 or 30%.

Question 5: Selecting a Student

In this question, we have a classroom with 20 boys and 15 girls. We need to find the probability of randomly selecting a boy.

Solution:

The classroom has a total of 20 + 15 = 35 students. Out of these, there are 20 boys. Therefore, the favorable outcomes are 20. Since there are 35 equally likely outcomes, the probability of randomly selecting a boy is 20/35, which simplifies to 4/7 or approximately 57.1%.

Question 6: Rolling Two Dice

This question involves rolling two fair six-sided dice and finding the probability of getting a sum greater than 8.

Solution:

When rolling two dice, there are a total of 6 x 6 = 36 equally likely outcomes. To find the favorable outcomes, we need to determine the combinations of numbers that result in a sum greater than 8. These combinations are (3, 6), (4, 5), (5, 4), (5, 5), (5, 6), (6, 3), (6, 4), (6, 5), and (6, 6). There are 9 favorable outcomes. Therefore, the probability of getting a sum greater than 8 when rolling two dice is 9/36 or 1/4, which simplifies to 25%.

Question 7: Drawing Cards with Replacement

This question involves drawing three cards with replacement from a standard deck of 52 playing cards and finding the probability of drawing three hearts.

Solution:

Since we are drawing with replacement, the probability of drawing a heart on each draw remains the same. From Question 2, we know that the probability of drawing a heart is 1/4. Since we are drawing three cards, we need to multiply the probabilities together. Therefore, the probability of drawing three hearts with replacement is (1/4) x (1/4) x (1/4) = 1/64 or approximately 1.6%.

Question 8: Selecting a Committee

In this question, there are 10 people, out of which 5 are women and 5 are men. We need to find the probability of randomly selecting a committee of 3 people with at least 2 women.

Solution:

To find the probability of selecting a committee with at least 2 women, we need to consider the different combinations. The favorable outcomes are selecting 2 women and 1 man, or selecting 3 women. The number of ways to select 2 women from 5 is given by the combination formula C(5, 2) = 10. The number of ways to select 1 man from 5 is given by the combination formula C(5, 1) = 5. The number of ways to select 3 women from 5 is given by the combination formula C(5, 3) = 10. Therefore, the total favorable outcomes are 10 + 5 + 10 = 25. The total number of possible outcomes is the combination formula C(10, 3) = 120. Therefore, the probability of randomly selecting a committee of 3 people with at least 2 women is 25/120 or approximately 20.8%.

Question 9: Drawing Balls from an Urn

This question involves drawing balls from an urn containing 4 red balls, 3 blue balls, and 2 green balls. We need to find the probability of drawing a blue ball followed by a green ball without replacement.

Solution:

The total number of balls in the urn is 4 + 3 + 2 = 9. To find the probability of drawing a blue ball followed by a green ball without replacement, we need to consider the favorable outcomes. The number of ways to select 1 blue ball from 3 is given by the combination formula C(3, 1) = 3. The number of ways to select 1 green ball from 2 is given by the combination formula C(2, 1) = 2. Therefore, the total favorable outcomes are 3 x 2 = 6. The total number of possible outcomes is the combination formula C(9, 2) = 36. Therefore, the probability of drawing a blue ball followed by a green ball without replacement is 6/36 or 1/6, which simplifies to approximately 16.7%.

Question 10: Spinning a Spinner

In this question, we are asked to find the probability of spinning a spinner and landing on a green section.

Solution:

The spinner has a total of 8 sections, out of which 2 are green. Therefore, the favorable outcomes are 2. Since there are 8 equally likely outcomes, the probability of landing on a green section is 2/8 or 1/4, which simplifies to 25%.

Question 11: Flipping Coins and Rolling a Die

This question involves flipping two coins and rolling a fair six-sided die. We need to find the probability of getting at least one tail and rolling an odd number.

Solution:

When rolling a die, the odd numbers are 1, 3, and 5. Therefore, the favorable outcomes for rolling an odd number are 3.

Since the coin flip and die roll are independent events, we need to multiply the probabilities together. Therefore, the probability of getting at least one tail and rolling an odd number is (3/4) x (3/6) = 9/24 or 37.5%.

Question 12: Selecting Cards from Different Decks

This question involves selecting a card from two different decks of playing cards and finding the probability of selecting a heart from the first deck and a diamond from the second deck.

Solution:

From Question 2, we