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What is the distribution of X? 1) Out of the 38 pizzas that you try, how many would you expect to like? 2) What is the variance of the number of pizzas that you like? Round to 3 d

What is the distribution of X? 1) Out of the 38 pizzas that you try, how many would you expect to like? 2) What is the variance of the number of pizzas that you like? Round to 3 d

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Only need Parts D and E

**Title: Understanding Probability and Decision Making with Pizza Choices** You don’t like pizza but all your friends love getting pizza so you always go with them and try a new topping and base sauce combination each time until you find a pizza that you like. The probability of you liking a pizza is 0.08. The average pizza costs $10 but your friends tell you that if you find a pizza that you like they will give you $80. --- **Problem (a):** Let \( Y \) be the number of times you try a new pizza before you find one you like. What kind of distribution does \( Y \) follow? What would be its expected value and variance? Don’t round your answers. **Solution:** - Distribution: \( Y \) follows a geometric distribution since we are counting the number of trials until the first success (finding a pizza you like). - Expected Value: \( E(Y) = \frac{1}{p} = \frac{1}{0.08} = 12.5 \) - Variance: \( \text{Var}(Y) = \frac{1-p}{p^2} = \frac{0.92}{(0.08)^2} = 143.75 \) --- **Problem (b):** You are determined to keep getting pizza until you find one that you like. When you finally find a pizza and your friends give you a congratulatory $80, will you have any profit? What will be the variance of this profit? **Solution:** - Cost to find a pizza you like: \( 10 \times E(Y) = 10 \times 12.5 = 125 \) - Profit: \( 80 - 125 = -45 \) (a loss of $45) - Variance of cost: \( 10^2 \times \text{Var}(Y) = 100 \times 143.75 = 14375 \) - As the profit is given by the reward minus the cost, the variance of the profit remains the same, i.e., 14375, assuming a fixed reward. --- **Problem (c):** You only start off with $60. What is the probability that you lose all of your money before you manage to find a pizza that you enjoy? Round your answer to 3 decimal places. **Solution:** The scenario describes running out of money before a success. Since each trial
Transcribed Image Text:**Title: Understanding Probability and Decision Making with Pizza Choices** You don’t like pizza but all your friends love getting pizza so you always go with them and try a new topping and base sauce combination each time until you find a pizza that you like. The probability of you liking a pizza is 0.08. The average pizza costs $10 but your friends tell you that if you find a pizza that you like they will give you $80. --- **Problem (a):** Let \( Y \) be the number of times you try a new pizza before you find one you like. What kind of distribution does \( Y \) follow? What would be its expected value and variance? Don’t round your answers. **Solution:** - Distribution: \( Y \) follows a geometric distribution since we are counting the number of trials until the first success (finding a pizza you like). - Expected Value: \( E(Y) = \frac{1}{p} = \frac{1}{0.08} = 12.5 \) - Variance: \( \text{Var}(Y) = \frac{1-p}{p^2} = \frac{0.92}{(0.08)^2} = 143.75 \) --- **Problem (b):** You are determined to keep getting pizza until you find one that you like. When you finally find a pizza and your friends give you a congratulatory $80, will you have any profit? What will be the variance of this profit? **Solution:** - Cost to find a pizza you like: \( 10 \times E(Y) = 10 \times 12.5 = 125 \) - Profit: \( 80 - 125 = -45 \) (a loss of $45) - Variance of cost: \( 10^2 \times \text{Var}(Y) = 100 \times 143.75 = 14375 \) - As the profit is given by the reward minus the cost, the variance of the profit remains the same, i.e., 14375, assuming a fixed reward. --- **Problem (c):** You only start off with $60. What is the probability that you lose all of your money before you manage to find a pizza that you enjoy? Round your answer to 3 decimal places. **Solution:** The scenario describes running out of money before a success. Since each trial
(d) Assume that you won a competition and now have a lifetime supply of pizzas for yourself for free. You go out for pizza 38 times in a year. Let \( X \) be the number of times you like your pizza out of the 38 different pizzas that you try. What is the distribution of \( X \)? (e) 1) Out of the 38 pizzas that you try, how many would you expect to like? 2) What is the variance of the number of pizzas that you like? Round to 3 decimal places.
Transcribed Image Text:(d) Assume that you won a competition and now have a lifetime supply of pizzas for yourself for free. You go out for pizza 38 times in a year. Let \( X \) be the number of times you like your pizza out of the 38 different pizzas that you try. What is the distribution of \( X \)? (e) 1) Out of the 38 pizzas that you try, how many would you expect to like? 2) What is the variance of the number of pizzas that you like? Round to 3 decimal places.
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