Question

Suppose that \(65 \%\) of all registered voters in a certain area favor a seven- day waiting period before purchase of a handgun. Among 225 randomly selected registered voters, what is the probability that a. At least 150 favor such a waiting period? b. More than 150 favor such a waiting period? c. Fewer than 125 favor such a waiting period?

Short answer

Exact answers for this situation may require a calculator or statistical software for dealing with binomial distribution function. However, we can say that for Case (a) the probability will likely be high, for Case (b) a bit lower, and Case (c) the lowest, given that each condition is less likely as per our original probability of 0.65.

Step-by-step solution

Define the Problem Variables

Let's define the variables for our binomial distribution: the number of trials \(n=225\) (the number of voters); the number of successes \(k\) (favoring the waiting period - this will vary depending on the case); and the probability of success \(p=0.65\).

Applying the Binomial Probability Formula

The binomial probability formula is: \(P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}\). However, we are interested in cumulative probability, not the specific probability for \(k\) so we will need to apply the cumulative binomial probability formula: \(P(X \geq k) = 1 - P(X < k)\). We will use these formulas to solve each case.

Calculate for Case (a)

Case (a) asks for the probability that at least 150 voters favor the waiting period. This means \(k \geq 150\). Using the cumulative binomial formula, we find \(P(X \geq 150) = 1 - P(X < 150)\). For the binomial cumulative distribution, you use statistical software or calculator with cumulative distribution function (CDF) but another way is to sum the probabilities from \(k = 0\) to \(149\).

Calculate for Case (b)

In case (b) we need to find the probability that more than 150 voters favor the waiting period, which means \(k > 150\). Similar to step 3, you use cumulative binomial distribution formula but this time calculate for \(k \geq 151\) which is \(P(X > 150) = 1 - P(X \leq 150)\). Again, calculate the cumulative sum up to \(150\).

Calculate for Case (c)

Case (c) is asking the probability that fewer than 125 voters favor the waiting period, implying \(k < 125\). In this situation, simply use the cumulative distribution function directly \(P(X < 125)\) to calculate the sum of probabilities from \(k = 0\) to \(124\).

Key concepts

Cumulative Binomial Probability

Understanding cumulative binomial probability helps in determining the likelihood of a given number of successes in several independent trials. It is essentially the sum of probabilities of all outcomes up to a certain number. Imagine tossing a coin: each toss is an independent trial, and getting a head could be considered a success. If we wanted to know the probability of getting at least one head in three tosses, we’d add up the probabilities of getting one, two, or three heads.

In the given exercise, a cumulative approach is used to answer questions about voters' preferences on a policy. Case (a) requires the cumulative probability of at least 150 voters in favor, which means all scenarios from 150 up to the total number, 225. Instead of calculating each scenario individually, the cumulative probability uses the logic of 'total probability minus the probability of all undesired outcomes' which is mathematically expressed as \(P(X \geq k) = 1 - P(X < k)\).

Calculating Cumulative Probability

In scenarios where manual calculation is required, you would add up the individual binomial probabilities from the lowest to just before the desired success count. However, due to the large number of trials in this exercise, using statistical software or a graphing calculator with a cumulative distribution function (CDF) feature is the practical approach.

Probability of Success in Binomial Distribution

The probability of success in a binomial distribution represents the chance that an individual trial results in a success. For our exercise, this is the probability that a randomly selected voter would favor the waiting period for handgun purchases, given as 65%.

This figure is central to calculating the overall likelihood of a specific number of people favoring the policy in a group of trials. By understanding the individual probability of success, we can better predict trends and make inferences about larger populations based on sample data.

Role in Binomial Distributions

In binomial distributions, the probability of success (denoted as \( p \)) does not change from trial to trial, and each trial is independent. This is a key feature of the binomial distribution— the consistency of \( p \) across all trials helps to maintain the distribution's defined structure and allows for the use of the binomial probability formula.

Binomial Probability Formula

The binomial probability formula is the core equation used to calculate the likelihood of observing a specific number of successes in a set number of trials. It integrates the number of trials, the number of successes, and the probability of success.

The formula is usually presented as: \[ P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} \] Where:\

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• \(n\) is the number of trials\
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• \(k\) is the number of successes\
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• \(p\) is the probability of success in a single trial\
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• \(1-p\) is the probability of failure in a single trial\
\

The formula makes use of the combination function, \( \binom{n}{k} \) which calculates the number of ways \( k \) successes can occur in \( n \) trials, considering the order of successes does not matter. This aspect of the formula reflects the different scenarios where the successes could occur throughout the trials.

Applying the Formula

In the exercise provided, the formula is applied to find probabilities for different counts of voters. For case (b) for example, calculating the probability for more than 150 voters needs an understanding of the cumulative probability. Although the formula gives the probability for an exact count, we use it to calculate probabilities up to a count, and then subtract from 1 to find the 'more than' or 'at least' probabilities. The binomial distribution is a powerful tool in statistics for modeling scenarios with two possible outcomes in each trial,a success or a failure.