Question

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Answer 1

We will investigate the use of Venn diagrams to determine the respective probabilities.

A Algebra 2 class consists of total students ( T ):

$T\text{ = 29 students}$

We will define two sports events ( A ) and ( B ) as follows:

$\begin{gathered} A\text{ : Students who play basketball} \\ B\text{ : Students who play baseball} \end{gathered}$

We will define the respective proportions of students associated with each event using set notations.

$\begin{gathered} A\text{ = 17 , B = 6} \\ p(A)=\frac{17}{29}\text{ , p ( B ) = }\frac{6}{29} \end{gathered}$$A^{\prime}\text{ \& B' = 8 students}$$p\text{ ( A' \& B' ) = }\frac{8}{29}$

We will construct a venn Diagram to grasp the entire distribution of events in the Algebra class:

From the above venn diagram and using the law of probabilities i.e all probabilities concerning a universal set must add up to 1.

$\text{Sum of all probabilities = 1}$$p\text{ ( A ) + p ( B ) - p ( A \& B ) + p ( A' \& B' ) = 1}$

We will use the above universal law of probabilities to determine the probability that a randomly chosen student from the class plays both basketball and baseball ( A & B ):

$\begin{gathered} \frac{17}{29}\text{ + }\frac{6}{29}\text{ + }\frac{8}{29}\text{ - 1 = p ( A \& B )} \\ \\ \text{p ( A \& B ) = }\frac{31}{29}\text{ - 1} \\ \\ \text{p ( A \& B ) = }\frac{31\text{ - 29}}{29}\text{ = }\frac{2}{29} \\ \\ \text{p ( A \& B ) = }\frac{2}{29}\text{ = 0.069 }\ldots\text{ Answer} \end{gathered}$