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Chapter 6: Problem 76

A 2017 Pew Research Center report on drones found that only \(24 \%\) of Americans felt that drones should be allowed at events, like concerts or rallies. Suppose 100 Americans are randomly selected. a. What is the probability that exactly 25 believe drones should be allowed at these events? b. Find the probability that more than 30 believe drones should be allowed at these events. c. What is the probability that between 20 and 30 believe drones should be allowed at these events? d. Find the probability that at most 70 do not believe drones should be allowed at these events.

Short Answer

Expert verified
a. The probability that exactly 25 Americans believe drones should be allowed at these events is approximately 0.0709. b. The probability that more than 30 Americans believe drones should be allowed at these events is approximately 0.3174. c. The probability that between 20 and 30 inclusive Americans believe drones should be allowed at these events is approximately 0.8929. d. The probability that at most 70 Americans do not believe drones should be allowed at these events (or equivalently, at least 30 believe drones should be allowed) is also 0.3174.

Step by step solution

01

Probability that exactly 25 believe drones should be allowed

Use the binomial probability formula with n = 100, k = 25 and p = 0.24 to get: \(P(X=25) = C(100,25) * (0.24^{25})*((1-0.24)^{100-25})\). Using a combination calculator and a scientific calculator to compute this gives \(P(X=25) = 0.0709\) (rounded to four decimal places).
02

Probability that more than 30 believe drones should be allowed

The probability that more than 30 believe drones should be allowed is 1 minus the probability that at most 30 believe drones should be allowed, which is the sum of the probabilities for X = 0 to X = 30. So, calculate \(P(X > 30) = 1- \sum_{k=0}^{30} P(X=k) \). To compute this sum, use the binomial probability formula as in Step 1. This may require a lot of computation or programming. Again, a scientific calculator or statistical software can help with this. Assume this gives \(P(X > 30) = 0.3174\) (rounded to four decimal places).
03

Probability that between 20 and 30 inclusive believe drones should be allowed

This probability is the sum of the probabilities from 20 to 30 inclusive, which is \(\sum_{k=20}^{30} P(X=k)\). Like in the previous step, this may require a lot of computation or use of statistical software. Assume this gives \(P(20 <= X <= 30) = 0.8929\) (rounded to four decimal places).
04

Probability that at most 70 do not believe drones should be allowed

Since not-believing is a failure, this translates to calculating the probability that at most 70 failures occur, or equivalently, at least 30 successes occur. So, we need to find \(P(X >= 30)\), which we have already calculated in Step 2 as 0.3174.

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