Question

Each student in a certain high school was classified according to her year in school (freshman, sophomore, junior, or senior) and according to the number of times that she had visited a certain museum (never, once, or more than once). The proportions of students in the various classifications are given in the following table:

Never once More than once

than once

Freshmen 0.08 0.10 0.04

Sophomores 0.04 0.10 0.04

Juniors 0.04 0.20 0.09

Seniors 0.02 0.15 0.10

a. If a student selected at random from the high school is a junior, what is the probability that she has never visited the museum?

b. If a student selected at random from the high school has visited the museum three times, what is the probability that she is a senior?

Short answer

1. Probability of student that she never visited the museum is \(\frac{4}{{33}}\)
2. Probability that a student visited museum more than three time she is a senior is \(\frac{{10}}{{27}}\)

The probability that at least one person receives exactly two aces in five cards is 0.1194.

Step-by-step solution

Given information

Students are classified into 4 categories-freshmen, sophomore, junior, or senior and according to the no. of times she visited to the museum

Computing the probability

a)

Probability of junior:

\(\begin{aligned}{\rm P}\left( {junior} \right) = 0.04 + 0.20 + 0.09\\ = 0.33\end{aligned}\)

Therefore

\(\begin{aligned}{\rm P}\left( {Never|Junior} \right) = \frac{{{\rm P}\left( {\,Junior\,and\,Never\,} \right)}}{{{\rm P}\left( {Junior} \right)}}\\ = \frac{{0.04}}{{0.33}}\\ = \frac{4}{{33}}\end{aligned}\)

State the simple events

b)

When the student visited museum three times, it means the student has visited the museum more than once.

So, let A be the event of senior visited museum more than once.

B be the event of senior and more than once

C be the event of visiting more than once

Therefore,

\(\begin{aligned}{\rm P}\left( {\rm A} \right) = \frac{{{\rm P}\left( B \right)}}{{{\rm P}\left( C \right)}}\\ = \frac{{0.10}}{{0.27}}\\ = \frac{{10}}{{27}}\end{aligned}\)