91Ó°ÊÓ

Probability is a fundamental concept in statistics that quantifies the likelihood of a specific event occurring. In the context of this exercise, a binomial distribution is used to model the situation where there are fixed numbers of trials or events (in this case, parolees), and each event has two possible outcomes: success (not becoming a repeat offender) or failure (becoming a repeat offender). The probability of success, denoted by \( p \), is given as \( 0.75 \), indicating that 75% of parolees do not re-offend.

The probability mass function (PMF) of a binomial distribution allows us to calculate the probability of exactly \( r \) successes in \( n \) trials, and is expressed by the formula:

\[P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}\]

In this exercise, mathematically \( n = 4 \) and \( r \) can range from 0 to 4. For instance, the probability that exactly two parolees are successes is calculated as \( P(2) = 0.1758 \), following the binomial probability formula.

The expected value is a key concept in probability that represents the average outcome if an experiment is repeated many times. It gives us a sense of the 'center' of the probability distribution.

• The formula for the expected value \( E(X) \) of a binomial distribution is \( E(X) = np \).
• In our exercise, with \( n = 4 \) parolees and probability of success \( p = 0.75 \), the expected value is \( E(X) = 4 \times 0.75 = 3 \).

This result implies that on average, Alice can expect 3 out of the 4 parolees not to become repeat offenders when considering similar situations multiple times.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In probability distributions, it tells us how much the values of the random variable differ from the expected value on average.

For a binomial distribution, the standard deviation \( \sigma \) is calculated using the formula:

\[\sigma = \sqrt{np(1-p)}\]

In the given exercise, with \( n = 4 \) and \( p = 0.75 \), the standard deviation is \( \sigma = \sqrt{4 \times 0.75 \times 0.25} = 0.866 \).

• This value helps Alice understand the variability in the number of parolees who are likely not to re-offend.
• A smaller standard deviation would indicate less variability, while a larger one would suggest more variability in the outcomes.

Histograms are a graphical representation of the distribution of numerical data. They are particularly useful for visualizing the shape of the binomial distribution in this exercise. A histogram displays the probability for each possible number of successes.

To create a histogram for this situation:

• The x-axis represents the number of parolees who are not repeat offenders (0 through 4).
• The y-axis represents the probabilities associated with these values.

Plotting the probabilities calculated (e.g., \( P(0) = 0.0039 \), \( P(1) = 0.0469 \), etc.) will show bars at each value of \( r \) with heights corresponding to these probabilities. This visual tool helps quickly assess where the most likely outcomes lie, enhancing the understanding of the distribution of successes in Alice's group.