Question

The student council for a school of science and math has one representative from each of five academic departments: Biology (B), Chemistry (C), Mathematics (M), Physics (P), and Statistics (S). Two of these students are to be randomly selected for inclusion on a university-wide student committee. a. What are the 10 possible outcomes? b. From the description of the selection process, all outcomes are equally likely. What is the probability of each outcome? c. What is the probability that one of the committee members is the statistics department representative? d. What is the probability that both committee members come from laboratory science departments?

Short answer

a. The 10 possible outcomes. b. The probability of each outcome is \(\frac{1}{10}\). c. The probability of one member being the statistics representative is 0.4. d. The probability of both members from laboratory departments is 0.3.

Step-by-step solution

Determine possible outcomes

To solve this, combinations formula can be used. In a combination, the order does not matter. There are 5 departments and 2 representatives need to be chosen. The formula for combinations is \(C(n, r) = \frac{n!}{r!(n-r)!}\) where n is the total number of options (5 departments) and r is the number of items to choose (2 representatives). This formula gives 10 possible outcomes.

Calculate the probability of each outcome

Since the two students are to be randomly selected, and there is no indication that there is any weighting or bias towards any particular department, each of the possible combinations is equally likely. Therefore, the probability of any specific outcome is thus \(\frac{1}{10}\).

Calculate the probability of the statistics representative being selected

There are 4 other departments to choose the second representative from. Which gives 4 possible outcomes when a statistics representative is included, which gives the probability \(\frac{4}{10} = 0.4\).

Calculate the probability that both members are from laboratory science departments

The laboratory science departments are Biology, Chemistry and Physics. We assume the order doesn't matter, so we find the combinations of 2 committee members from 3 departments by using the combinations formula as in Step 1 to get 3 possible outcomes. This gives the probability \(\frac{3}{10}= 0.3\).