91Ó°ÊÓ

Let 'A' be the event in which a randomly selected museum utilizes total visitors as a performance measure the most often, and 'B' be the event in which funds raised are used as a performance measure. The probability that the museum utilizes total visitors or funds raised as a performance measure is:

$\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{∪}\mathbf{B}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{-}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{âˆ\copyright }\mathbf{B}\mathbf{)}$

As the Number of museums using total visitors as a performance measure = 8, So:

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\frac{\mathbf{8}}{\mathbf{30}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{4}}{\mathbf{15}}\end{array}$

As some museums use funds raised as performance measure = 7, So:

$\mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{=}\frac{\mathbf{7}}{\mathbf{30}}$

Therefore,

$\begin{array}{c}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{∪}\mathbf{B}\mathbf{)}\mathbf{=}\frac{\mathbf{4}}{\mathbf{15}}\mathbf{+}\frac{\mathbf{7}}{\mathbf{30}}\mathbf{−}\mathbf{0}\\ \mathbf{}\mathbf{=}\frac{\mathbf{8}\mathbf{+}\mathbf{7}}{\mathbf{30}}\\ \mathbf{=}\frac{\mathbf{15}}{\mathbf{30}}\\ \mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\end{array}$

$\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{âˆ\copyright }\mathbf{B}\mathbf{)}$is the probability that the museum utilizes both total visitors and funds raised as performance measures cannot be done continuously.

Hence, $\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{âˆ\copyright }\mathbf{B}\mathbf{)}\mathbf{=}\mathbf{0}$.

As the tree diagram demonstrates, the probability is as follows:

Because the two museums can utilize any performance measure, they are completely independent. Therefore, the chances of either museum selecting any of the five measures remain the same.

Hence, the sample points' probabilities are as follows:

$\begin{array}{l}\mathbf{Museum}\mathbf{1}\mathbf{.}\\ \mathbf{Total}\mathbf{}\mathbf{visitors}\mathbf{=}\frac{\mathbf{8}}{\mathbf{30}}\mathbf{x}\frac{\mathbf{8}}{\mathbf{30}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{64}}{\mathbf{900}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{16}}{\mathbf{225}}\end{array}$

$\begin{array}{l}\mathbf{Fundvisitors}\mathbf{=}\frac{\mathbf{8}}{\mathbf{30}}\mathbf{x}\frac{\mathbf{7}}{\mathbf{30}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{56}}{\mathbf{900}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{28}}{\mathbf{450}}\end{array}$

$\begin{array}{l}\mathbf{Museum}\mathbf{2}\mathbf{.}\\ \mathbf{Totalvisitors}\mathbf{=}\frac{\mathbf{9}}{\mathbf{30}}\mathbf{x}\frac{\mathbf{8}}{\mathbf{30}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{72}}{\mathbf{900}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{14}}{\mathbf{225}}\end{array}$

$\begin{array}{l}\mathbf{Fund}\mathbf{}\mathbf{visitors}\mathbf{=}\frac{\mathbf{7}}{\mathbf{30}}\mathbf{x}\frac{\mathbf{7}}{\mathbf{30}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{64}}{\mathbf{900}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{49}}{\mathbf{900}}\end{array}$

Hence, the probabilities of museums 1 & 2 are 16/225, 28/450, 14/225, and 49/900.

$$\begin{gathered} \mathbf{P}\mathbf{=}\frac{\mathbf{16}}{\mathbf{225}}\mathbf{+}\frac{\mathbf{28}}{\mathbf{450}}\mathbf{+}\frac{\mathbf{14}}{\mathbf{225}}\mathbf{+}\frac{\mathbf{49}}{\mathbf{900}} \\ \mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{64}\mathbf{+}\mathbf{56}\mathbf{+}\mathbf{56}\mathbf{+}\mathbf{49}}{\mathbf{900}} \\ \mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{225}}{\mathbf{900}} \\ \mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}} \end{gathered}$$

Hence, the probability of both the museum is 1/4.