Health The table shows the probability distribution of the...

Health The table shows the probability distribution of the numbers of AIDS cases diagnosed in the United States in 2005 by age group. $$ \begin{array}{l}{\text { (a) Sketch the probability distribution. }} \\ {\text { (b) Find the probability that an individual diagnosed with }} \\ {\text { AIDS was from } 15 \text { to } 44 \text { years of age. }} \\ {\text { (c) Find the probability that an individual diagnosed with }} \\ {\text { AIDS was at least } 35 \text { years of age. }} \\ {\text { (d) Find the probability that an individual diagnosed with }} \\ {\text { AIDS was at most } 24 \text { years of age. }}\end{array} $$

Health The table shows the probability distribution of the numbers of AIDS cases diagnosed in the United States in 2005 by age group.
$$
\begin{array}{l}{\text { (a) Sketch the probability distribution. }} \\ {\text { (b) Find the probability that an individual diagnosed with }} \\ {\text { AIDS was from } 15 \text { to } 44 \text { years of age. }} \\ {\text { (c) Find the probability that an individual diagnosed with }} \\ {\text { AIDS was at least } 35 \text { years of age. }} \\ {\text { (d) Find the probability that an individual diagnosed with }} \\ {\text { AIDS was at most } 24 \text { years of age. }}\end{array}
$$

Key Concepts

Probability Distribution

A probability distribution is a mathematical function that provides the probabilities of occurrence of different outcomes in an experiment. It shows how probabilities are distributed over the values of the random variable, ensuring that the sum of all probabilities is equal to one. In the context of this question, the distribution assigns probabilities to different age groups diagnosed with AIDS.

Graphical Representation

Graphical representation of a probability distribution, such as a histogram or a bar plot, is a visual tool that helps to understand the distribution of probabilities across outcomes. It makes it easier to identify patterns and trends by providing an immediate visual summary of the data.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes a value less than or equal to a certain threshold. It is calculated by summing the probabilities for all outcomes up to that value. For instance, finding the probability of a case being at most a certain age involves the cumulative probability up to that age.

Summation of Probabilities

Summing probabilities involves combining the probabilities of several mutually exclusive events to find the total probability that any one of those events occurs. This is fundamental when dealing with grouped or interval data, such as computing the probability of an event falling within a range of values.

Interval Probabilities

Interval probabilities calculate the likelihood that a random variable falls within a specific range. This concept is particularly important when outcomes are grouped into intervals, as one needs to sum the probabilities of all outcomes in the given interval to obtain the desired probability.

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