Problem 26 - Algebra and Trigonometry

Question

Political Surveys In a certain county, 60\(\%\) of the voters are in favor of an upcoming school bond initiative. If 5 voters are interviewed at random, what is the probability that exactly 3 of them will favor the initiative?

Step-by-Step Solution

Answer

The probability is approximately 34.56\%."

Step 1: Identify the Distribution Type

The problem describes a situation where we have a fixed number of independent trials, each with two possible outcomes (in favor or not in favor of the initiative), and we want to find the probability of a specific number of successes (voters in favor). This is a classic binomial distribution scenario. We have \( n = 5 \) trials (voters) and probability of success \( p = 0.6 \) (favoring the initiative).

Step 2: Use the Binomial Probability Formula

The probability of exactly \( k \) successes in \( n \) independent Bernoulli trials, each with success probability \( p \), is given by the binomial probability formula:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Substitute \( n = 5 \), \( k = 3 \), and \( p = 0.6 \) into the formula. This gives us:\[ P(X = 3) = \binom{5}{3} (0.6)^3 (0.4)^2 \]

Step 3: Calculate the Binomial Coefficient

The binomial coefficient \( \binom{n}{k} \) is calculated as \( \frac{n!}{k!(n-k)!} \). For \( \binom{5}{3} \), we calculate:\[ \binom{5}{3} = \frac{5!}{3!2!} = \frac{5 \times 4}{2 \times 1} = 10 \]

Step 4: Calculate the Powers

Now we calculate the powers of the probabilities:- \( (0.6)^3 = 0.6 \times 0.6 \times 0.6 = 0.216 \)- \( (0.4)^2 = 0.4 \times 0.4 = 0.16 \)

Step 5: Calculate the Probability

Substitute the binomial coefficient and powers back into the formula:\[ P(X = 3) = 10 \times 0.216 \times 0.16 \]Calculate the result:\[ = 10 \times 0.03456 = 0.3456 \]

Step 6: Conclusion

The probability that exactly 3 out of 5 randomly interviewed voters favor the school bond initiative is approximately \( 0.3456 \), or 34.56\( \% \).

Key Concepts

Understanding Probability

Probability helps us measure how likely an event is to occur. When we say the probability of an event is high, it means there is a good chance the event will happen. Conversely, a low probability suggests the event is less likely. In mathematics, probability is expressed as a number between 0 and 1. For example, a probability of 0 means the event will not happen, while 1 means it certainly will happen. To calculate the probability of a single event happening, you can use the formula:

• Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes)

In the context of binomial distribution, probability helps us find the chance of achieving a certain number of successes in a series of experiments. This is crucial to understanding scenarios where events have only two outcomes, like a yes or no situation.

The Binomial Probability Formula

The binomial probability formula is an essential tool in finding the probability of a specific number of successes in a series of Bernoulli trials. These trials are trials with only two possible outcomes: success or failure. The binomial probability formula is given by:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Here's what each part stands for:

• \( P(X = k) \) is the probability of getting exactly \( k \) successes.
• \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways \( k \) successes can occur in \( n \) trials.
• \( p \) is the probability of success on a single trial.
• \( (1-p) \) is the probability of failure on a single trial.
• \( n \) is the total number of trials.
• \( k \) is the number of successful trials you want to find the probability for.

Using this formula allows us to neatly calculate complex probability scenarios that involve multiple attempts or trials.

Understanding Bernoulli Trials

A Bernoulli trial is a random experiment where there are only two outcomes: success or failure. These trials are called 'Bernoulli' after the Swiss mathematician Jacob Bernoulli, who studied them in detail. When you're dealing with questions about likelihood that involve yes or no, true or false, or success or failure scenarios, you're dealing with Bernoulli trials.Some key traits of Bernoulli trials include:

• Independence: Each trial is independent; the result of one does not affect the others.
• Identical probability: The probability of success, \( p \), remains the same in each trial.

When circumstances fit the criteria of Bernoulli trials and you want to calculate the probability of a certain number of successes over several trials, that's when the binomial distribution and its associated formula become useful tools.

The Role of the Binomial Coefficient

The binomial coefficient is integral to calculating probabilities using the binomial probability formula. It provides the number of ways to choose \( k \) successes from \( n \) trials. The binomial coefficient is denoted as \( \binom{n}{k} \) and calculated using the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]Here's the breakdown:

• \( n! \) (n factorial) is the product of all positive integers up to \( n \).
• \( k! \) (k factorial) is the product of all positive integers up to \( k \).
• \( (n-k)! \) is the factorial of the difference between \( n \) and \( k \).

The binomial coefficient calculates the number of different ways to arrange successes and failures in the given number of trials. This concept is crucial, as it factors in all the possible outcomes and orientations of successes across trials—paving the way for accurately determining probability outcomes in binomial distributions.