USA Today reported that approximately 25 % of all state...

USA Today reported that approximately $25 \%$ of all state prison inmates released on parole become repeat offenders while on parole. Suppose the parole board is examining five prisoners up for parole. Let $x=$ number of prisoners out of five on parole who become repeat offenders. The methods of Section 5.2 can be used to compute the probability assignments for the $x$ distribution. $$\begin{array}{c|cccccc} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline P(x) & 0.237 & 0.396 & 0.264 & 0.088 & 0.015 & 0.001 \\ \hline \end{array}$$ (a) Find the probability that one or more of the five parolees will be repeat offenders. How does this number relate to the probability that none of the parolees will be repeat offenders? (b) Find the probability that two or more of the five parolees will be repeat offenders. (c) Find the probability that four or more of the five parolees will be repeat offenders. (d) Compute $\mu,$ the expected number of repeat offenders out of five. (e) Compute $\sigma,$ the standard deviation of the number of repeat offenders out of five.

USA Today reported that approximately $25 \%$ of all state prison inmates released on parole become repeat offenders while on parole. Suppose the parole board is examining five prisoners up for parole. Let $x=$ number of prisoners out of five on parole who become repeat offenders. The methods of Section 5.2 can be used to compute the probability assignments for the $x$ distribution.
$$\begin{array}{c|cccccc} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline P(x) & 0.237 & 0.396 & 0.264 & 0.088 & 0.015 & 0.001 \\ \hline \end{array}$$
(a) Find the probability that one or more of the five parolees will be repeat offenders. How does this number relate to the probability that none of the parolees will be repeat offenders?
(b) Find the probability that two or more of the five parolees will be repeat offenders.
(c) Find the probability that four or more of the five parolees will be repeat offenders.
(d) Compute $\mu,$ the expected number of repeat offenders out of five.
(e) Compute $\sigma,$ the standard deviation of the number of repeat offenders out of five.

Key Concepts

Expected Value

The expected value is the theoretical mean or average of a probability distribution, representing the long-run average outcome of repeated trials. In the context of a binomial distribution, the expected value is calculated as the product of the number of trials and the probability of success.

Binomial Distribution

This is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has two possible outcomes (success or failure) and a constant probability of success. It is used to model situations such as determining how many individuals out of a group will exhibit a particular trait.

Complement Rule

This rule states that the probability of an event occurring is equal to one minus the probability of it not occurring. It is especially useful for finding probabilities of 'at least' or 'one or more' events by calculating the simpler complimentary probability.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specified value, or alternatively, that it is at least a certain value. It involves adding up individual probabilities from the probability mass function to obtain the total probability over a range of outcomes.

Variance and Standard Deviation

The variance measures the spread of the distribution around the expected value, while the standard deviation is the square root of the variance and indicates typical deviation from the mean. For a binomial distribution, the variance is computed as the product of the number of trials, the probability of success, and the probability of failure.

Transcript

Please give Ace some feedback