Prison Inmates Forty percent of prison inmates were...

Prison Inmates Forty percent of prison inmates were unemployed when they entered prison. If 5 inmates are randomly selected, find these probabilities: $$ \begin{array}{l}{\text { a. Exactly } 3 \text { were unemployed. }} \\ {\text { b. At most } 4 \text { were unemployed. }} \\ {\text { c. At least } 3 \text { were unemployed. }} \\ {\text { d. Fewer than } 2 \text { were unemployed. }}\end{array} $$

Prison Inmates Forty percent of prison inmates were
unemployed when they entered prison. If 5 inmates are
randomly selected, find these probabilities:
$$
\begin{array}{l}{\text { a. Exactly } 3 \text { were unemployed. }} \\ {\text { b. At most } 4 \text { were unemployed. }} \\ {\text { c. At least } 3 \text { were unemployed. }} \\ {\text { d. Fewer than } 2 \text { were unemployed. }}\end{array}
$$

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Key Concepts

Combinations

Combinations are used to calculate the number of ways to choose a specific number of successes from a set number of trials. This concept, often denoted as 'n choose k', is essential in the binomial formula to account for all possible arrangements of successes and failures.

Cumulative Probabilities

This concept involves summing the probabilities of several outcomes to satisfy conditions such as 'at most', 'at least', or 'fewer than' a specified number of successes. It is important for evaluating probabilities over a range of outcomes rather than a single specific event.

Binomial Distribution

This concept models scenarios with a fixed number of independent trials, each of which results in a binary outcome (success or failure). It is used when calculating the probability of achieving a specific number of successes given a constant probability of success in each trial.

Probability of Success

In the context of the binomial distribution, the probability of success remains the same in each independent trial. Understanding this concept is crucial for determining the likelihood of a single event occurring and subsequently applying it in repeated trials.

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