Problem 30 - Algebra 2

Question

In a recent year, it was estimated that 55\(\%\) of U.S. adult Internet users had access to high-speed Internet connections at home or on the job. What is the probability that exactly 2 out of 5 randomly selected U.S. adults had access to high-speed Internet connections?

Step-by-Step Solution

Answer

The probability is 0.275.

Step 1: Identify the Type of Problem

This problem is a probability problem dealing with binomial probability distribution where we want to find the probability of exactly 2 successes (having access to high-speed Internet) in 5 trials.

Step 2: Define Parameters for Binomial Distribution

Identify and define the parameters for the binomial distribution: the number of trials \( n = 5 \), the number of successes \( k = 2 \), and the success probability \( p = 0.55 \). The failure probability \( q = 1 - p = 0.45 \).

Step 3: Write the Binomial Probability Formula

The formula to calculate binomial probability is \( P(X = k) = \binom{n}{k} p^k q^{n-k} \). Here \( \binom{n}{k} \) is the binomial coefficient \( \frac{n!}{k!(n-k)!} \).

Step 4: Calculate the Binomial Coefficient

Calculate the binomial coefficient for \( \binom{5}{2} \): \[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \].

Step 5: Calculate the Probability Component

Calculate the probability components: \( p^2 = 0.55^2 = 0.3025 \), and \( q^{5-2} = 0.45^3 = 0.091125 \).

Step 6: Apply the Binomial Formula

Now substitute all values into the binomial probability formula: \[ P(X = 2) = 10 \times 0.3025 \times 0.091125 \].Calculate it to get the probability.

Step 7: Calculate Result

Perform the multiplication: \( 10 \times 0.3025 \times 0.091125 = 0.275 \). So, the probability that exactly 2 out of 5 randomly selected U.S. adults have access to high-speed Internet is 0.275.

Key Concepts

Probability

When we talk about probability, we are referring to the likelihood or chance of an event happening. In the context of this problem, we're assessing the probability that a specific number of people, out of a larger group, fit a certain criteria—having access to high-speed Internet, for instance.

• Probability is usually represented as a number between 0 and 1.
• A probability of 0 means the event will not occur, whereas a probability of 1 means the event is certain to occur.

In our exercise, we calculate the probability of exactly two out of five selected adults having high-speed internet access.
This involves determining what the chances are for a specific outcome in a series of trials.

Binomial Coefficient

The binomial coefficient is a critical part of working with binomial probabilities. It denotes the number of ways to choose a certain number of successes within a set number of trials.

• The mathematical notation for binomial coefficient is \( \binom{n}{k} \), which is read as "n choose k."
• The formula for the binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

Here, \( n \) is the total number of trials, and \( k \) is the number of successes you are interested in finding.
For our problem, we calculated \( \binom{5}{2} = \frac{5!}{2!(5-2)!} \), which equals 10, representing the 10 different combinations of having exactly 2 successes in 5 trials.

Binomial Formula

The binomial formula provides a way to calculate the probability of achieving exactly \( k \) successes in \( n \) trials of an experiment. The formula is:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]In this formula:

• \( \binom{n}{k} \) is the binomial coefficient, which provides the number of ways to achieve \( k \) successes.
• \( p^k \) represents the probability of achieving \( k \) successes.
• \((1-p)^{n-k} \) computes the probability of the remaining \( n-k \) trials being failures.

For our exercise, this formula helps us calculate that there is a 0.275 probability of exactly 2 out of 5 adults having access to high-speed Internet.

Success Probability

Success probability is a straightforward yet vital aspect of the binomial probability distribution. It explains how likely it is that each individual trial or event will be a success.

• This probability is represented by \( p \), with a value ranging from 0 to 1.
• In our scenario, \( p = 0.55 \), meaning there is a 55% chance that any given adult has high-speed Internet access.

It's crucial to also understand the complementary probability, known as failure probability or \( q \), which is calculated by \( q = 1 - p \).
For our example, since \( p = 0.55 \), then \( q = 0.45 \) or a 45% chance that a given adult does not have high-speed Internet.
Knowing both the success and failure probabilities is important because they are plugged into the binomial formula to solve problems like these.