6. Stat A. Stishen is a Business Statistics student at State U. Unfortunately, he is not a very mature student, and he shows up for the final exam having spent no time preparing for it. During the multiple choice (MC) portion of the exam, he simply uses the random integer command on his calculator to make guesses for each item. Each item has five possible choices (only one is correct). The items referenced below only refer to this portion of the exam. (a) What is the probability Stat's first correct response is on the third item? (b) What is the probability it takes at least 8 items for Stat's first correct response? (c) How many items do we expect Stat to answer before his first correct response? Interpret the result
6. Stat A. Stishen is a Business Statistics student at State U. Unfortunately, he is not a very mature student, and he shows up for the final exam having spent no time preparing for it. During the multiple choice (MC) portion of the exam, he simply uses the random integer command on his calculator to make guesses for each item. Each item has five possible choices (only one is correct). The items referenced below only refer to this portion of the exam. (a) What is the probability Stat's first correct response is on the third item? (b) What is the probability it takes at least 8 items for Stat's first correct response? (c) How many items do we expect Stat to answer before his first correct response? Interpret the result
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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Question 6.0 - Bernoulli Trials
binompdf and binomcdf (calculator)
Please solve the problem with simple probability rules
![### Probability Concepts in Multiple Choice Exams
**Example Scenario:**
Stat A. Stishen is a Business Statistics student at State U. Unfortunately, he is not a very mature student and he shows up for the final exam having spent no time preparing for it. During the multiple-choice (MC) portion of the exam, he simply uses the random integer command on his calculator to make guesses for each item. Each item has five possible choices (only one is correct). The items referenced below only refer to this portion of the exam.
**Questions:**
(a) **What is the probability that Stat’s first correct response is on the third item?**
To solve this, we assume Stat guesses randomly for each item with each having an equal probability. Since each item has five choices, the probability of guessing correctly is:
\[ P(\text{Correct}) = \frac{1}{5} \]
Conversely, the probability of guessing incorrectly is:
\[ P(\text{Incorrect}) = \frac{4}{5} \]
For the first correct response to be on the third item, Stat must guess incorrectly on the first two items and correctly on the third:
\[ P( \text{First correct on third item} ) = \left( \frac{4}{5} \right)^2 \times \left( \frac{1}{5} \right) \]
So:
\[ P( \text{First correct on third item} ) = \frac{16}{25} \times \frac{1}{5} = \frac{16}{125} = 0.128 \]
(b) **What is the probability it takes at least 8 items for Stat’s first correct response?**
This implies Stat guesses incorrectly the first 7 times, then either guesses correctly on the 8th item or even later. The probability of guessing incorrectly 7 times consecutively is:
\[ P(\text{Incorrect 7 times}) = \left( \frac{4}{5} \right)^7 \]
So:
\[ P(\text{At least 8 items}) = \left( \frac{4}{5} \right)^7 \]
Calculating this:
\[ P(\text{At least 8 items}) = \left( \frac{4}{5} \right)^7 \approx 0.209 \]
(c) **How many items do we expect Stat to answer before his first correct response? Interpret](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57c1a9e8-096b-4fde-a3f9-c0ea42610dcd%2F25000b19-a7ad-4804-8aa5-64681a639582%2Fv7swlj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Probability Concepts in Multiple Choice Exams
**Example Scenario:**
Stat A. Stishen is a Business Statistics student at State U. Unfortunately, he is not a very mature student and he shows up for the final exam having spent no time preparing for it. During the multiple-choice (MC) portion of the exam, he simply uses the random integer command on his calculator to make guesses for each item. Each item has five possible choices (only one is correct). The items referenced below only refer to this portion of the exam.
**Questions:**
(a) **What is the probability that Stat’s first correct response is on the third item?**
To solve this, we assume Stat guesses randomly for each item with each having an equal probability. Since each item has five choices, the probability of guessing correctly is:
\[ P(\text{Correct}) = \frac{1}{5} \]
Conversely, the probability of guessing incorrectly is:
\[ P(\text{Incorrect}) = \frac{4}{5} \]
For the first correct response to be on the third item, Stat must guess incorrectly on the first two items and correctly on the third:
\[ P( \text{First correct on third item} ) = \left( \frac{4}{5} \right)^2 \times \left( \frac{1}{5} \right) \]
So:
\[ P( \text{First correct on third item} ) = \frac{16}{25} \times \frac{1}{5} = \frac{16}{125} = 0.128 \]
(b) **What is the probability it takes at least 8 items for Stat’s first correct response?**
This implies Stat guesses incorrectly the first 7 times, then either guesses correctly on the 8th item or even later. The probability of guessing incorrectly 7 times consecutively is:
\[ P(\text{Incorrect 7 times}) = \left( \frac{4}{5} \right)^7 \]
So:
\[ P(\text{At least 8 items}) = \left( \frac{4}{5} \right)^7 \]
Calculating this:
\[ P(\text{At least 8 items}) = \left( \frac{4}{5} \right)^7 \approx 0.209 \]
(c) **How many items do we expect Stat to answer before his first correct response? Interpret
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