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Chapter 5: Problem 58

According to a Harris Interactive poll, \(52 \%\) of American college graduates have Facebook accounts (http://www.harrisinteractive.com/harris_poll/pubs/Harris_Poll \(200904_{-16}\).pdf). Suppose that this result. is true for the current population of American college graduates. a. Let \(x\) be a binomial random variable that denotes the number of American college graduates in a random sample of 15 who have Facebook accounts. What are the possible values that \(x\) can assume? b. Find the probability that exactly 9 American college graduates in a sample of 15 have Facebook accounts.

Short Answer

Expert verified
a. The possible values that \(x\) can assume are whole numbers from 0 to 15. b. The probability that exactly 9 American college graduates in a sample of 15 have Facebook accounts is found by evaluating the binomial probability formula.

Step by step solution

01

Determine the possible values

In a binomial distribution, \(x\) can assume all values from 0 to \(n\) where \(n\) is the number of trials. Here, \(n = 15\). Hence, \(x\) could be any whole number from 0 to 15.
02

Determine the probability

The probability \(P(X=x)\) of getting exactly \(x\) successes in \(n\) trials is given by the formula for the binomial probability: \[P(X=x) = \binom{n}{x} * (p^x) * ((1-p)^{n-x})\]Here, \(\binom{n}{x}\) represents the number of ways to choose \(x\) successes from \(n\) trials, \(p^x\) is the probability of \(x\) successes and \((1-p)^{n-x}\) is the probability of \(n-x\) failures. Apply this formula by substituting \(n = 15, x = 9\) and \(p = 0.52\) (52%).
03

Compute the combination

First we need to calculate \(\binom{n}{x}\) which is given as\[\binom{n}{x} = \frac{n!}{x!(n-x)!}\]by substituting \(n = 15\) and \(x = 9\).
04

Compute the probabilities

Next, compute \(p^x\) and \((1-p)^{n-x}\) by substituting \(p = 0.52, x = 9\) and \(n = 15\).
05

Multiply the terms

Multiply the results from Step 3 and Step 4 to get the final probability. This is the required probability that exactly 9 American college graduates in a sample of 15 have Facebook accounts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculation
Probability calculation is a fundamental concept in statistics that helps us measure how likely an event is to occur. In problems involving binomial distribution, probability calculation allows us to determine the likelihood of a specific number of successes in a series of independent trials.
To calculate probabilities in a binomial setting, we use the binomial probability formula. This formula requires you to know:
  • The total number of trials (\( n \)): This is the number of times an experiment or action is carried out. In our exercise, it's the number of college graduates sampled, which is 15.
  • The probability of one success in a single trial (\( p \)): This is the likelihood of an individual event being successful鈥攈ere, it's the probability that a graduate has a Facebook account, given as 52% or 0.52.
  • The number of successes you're interested in (\( x \)): This is the specific event count you want to determine the probability for鈥攊n this case, 9 graduates having Facebook accounts.
Using these values in the binomial probability formula helps to compute the probability of exactly \( x \) successes.
Random Variable
A random variable is a variable whose possible values are outcomes of a random phenomenon. In the context of the binomial distribution, we often define a random variable that denotes the number of successes in a fixed number of independent trials.
For this exercise, let \( x \) be the random variable representing the number of American college graduates with Facebook accounts in a sample of 15.
The possible values that this random variable can assume are any whole number from 0 to 15.
This range reflects that in a sample of 15 graduates, anywhere from none to all can potentially have a Facebook account. Each possible value corresponds to a different potential outcome of the sampling process.
Binomial Probability Formula
The binomial probability formula is critical for solving binomial distribution problems. It allows us to calculate the probability of a random variable assuming a specific value.
The formula is expressed as:
\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot (1-p)^{n-x} \]
Here's what each part means:
  • \( \binom{n}{x} \): The number of combinations, or ways, to choose \( x \) successes out of \( n \) trials. It's calculated using the formula:
    \[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \]
  • \( p^x \): The probability of achieving exactly \( x \) successes assuming each individual trial has a success probability \( p \).
  • \( (1-p)^{n-x} \): The probability that the remaining trials (\( n-x \)) result in failures.
In our exercise example, to find the probability of exactly 9 out of 15 graduates having Facebook accounts, plug in \( n = 15 \), \( x = 9 \), and \( p = 0.52 \) into the formula. These calculations will yield the probability specifically sought in such scenarios.

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