to buy drugs. P(X=2)= (Round to three decimal places as needed.) (b) Find the probability that among the next 5 theft cases reported in this district, at most 3 resulted from th drugs. P(X≤3) = (Round to three decimal places as needed.)
to buy drugs. P(X=2)= (Round to three decimal places as needed.) (b) Find the probability that among the next 5 theft cases reported in this district, at most 3 resulted from th drugs. P(X≤3) = (Round to three decimal places as needed.)
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![### Probability and Statistics Problem
In a certain city district, the need for money to buy drugs is stated as the reason for 62% of all thefts. Find the probability that among the next 5 theft cases reported in this district,
(a) exactly 2 resulted from the need for money to buy drugs.
(b) at most 3 resulted from the need to buy drugs.
---
(a) Find the probability that among the next 5 theft cases reported in this district, exactly 2 resulted from the need for money to buy drugs.
\[ P(X = 2) = \_\_\_ \]
(Round to three decimal places as needed.)
(b) Find the probability that among the next 5 theft cases reported in this district, at most 3 resulted from the need to buy drugs.
\[ P(X \leq 3) = \_\_\_ \]
(Round to three decimal places as needed.)
---
### Explanation
**Understanding the Problem:**
- **Given:** The probability \( p = 0.62 \) that a theft is due to the need to buy drugs.
- **Required:** The probability of specific outcomes among a sample of 5 thefts (n = 5).
**Instructions:**
1. **(a) Probability for exactly 2 cases:**
Use the binomial probability formula to calculate \( P(X = 2) \):
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
where:
- \( \binom{n}{k} \) is the binomial coefficient
- \( n = 5 \)
- \( k = 2 \)
- \( p = 0.62 \)
2. **(b) Probability for at most 3 cases:**
Compute the cumulative binomial probability \( P(X \leq 3) \):
\[
P(X \leq 3) = \sum_{k=0}^{3} \binom{n}{k} p^k (1-p)^{n-k}
\]
Make sure to round the final answers to three decimal places as needed.
**Note:**
You can use statistical software or a binomial probability calculator to find the exact values accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d1e5150-1781-42a0-9e07-3d8cfcc0113b%2F573e5e22-9f33-4e99-9336-37316baaa897%2F530ei4_processed.png&w=3840&q=75)
Transcribed Image Text:### Probability and Statistics Problem
In a certain city district, the need for money to buy drugs is stated as the reason for 62% of all thefts. Find the probability that among the next 5 theft cases reported in this district,
(a) exactly 2 resulted from the need for money to buy drugs.
(b) at most 3 resulted from the need to buy drugs.
---
(a) Find the probability that among the next 5 theft cases reported in this district, exactly 2 resulted from the need for money to buy drugs.
\[ P(X = 2) = \_\_\_ \]
(Round to three decimal places as needed.)
(b) Find the probability that among the next 5 theft cases reported in this district, at most 3 resulted from the need to buy drugs.
\[ P(X \leq 3) = \_\_\_ \]
(Round to three decimal places as needed.)
---
### Explanation
**Understanding the Problem:**
- **Given:** The probability \( p = 0.62 \) that a theft is due to the need to buy drugs.
- **Required:** The probability of specific outcomes among a sample of 5 thefts (n = 5).
**Instructions:**
1. **(a) Probability for exactly 2 cases:**
Use the binomial probability formula to calculate \( P(X = 2) \):
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
where:
- \( \binom{n}{k} \) is the binomial coefficient
- \( n = 5 \)
- \( k = 2 \)
- \( p = 0.62 \)
2. **(b) Probability for at most 3 cases:**
Compute the cumulative binomial probability \( P(X \leq 3) \):
\[
P(X \leq 3) = \sum_{k=0}^{3} \binom{n}{k} p^k (1-p)^{n-k}
\]
Make sure to round the final answers to three decimal places as needed.
**Note:**
You can use statistical software or a binomial probability calculator to find the exact values accurately.
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