Question

The student council for a school of science and math has one representative from each of the 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 (by placing five slips of paper in a bowl, mixing, and drawing out two of them). a. What are the 10 possible outcomes (simple events)? b. From the description of the selection process, all outcomes are equally likely; what is the probability of each simple event? 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 are: 'BC', 'BM', 'BP', 'BS', 'CM', 'CP', 'CS', 'MP', 'MS', 'PS'. b. The probability of each simple event is 0.1. c. The probability that one of the committee members is the statistics department representative is 0.4. d. The probability that both committee members come from laboratory science departments is 0.6.

Step-by-step solution

Listing All Possible Outcomes

Identify all the possible combinations or outcomes of selecting 2 representatives from the 5 departments. Use the combination formula by which number of ways to choose 'k' from 'n' objects is denoted as \(C(n, k) = \frac{n!}{k!(n-k)!}\). For this case, \(C(5, 2) =10\), let's list out the 10 possible outcomes :\['BC', 'BM', 'BP', 'BS', 'CM', 'CP', 'CS', 'MP', 'MS', 'PS'\]

Determine The Probability Of Each Simple Event

Let A be the set of all possible outcomes then the probability of A happening is 1. The probability of each case is equal, hence each individual pair has a 1/10 chance of being chosen.

Calculating the probability of having a representative from the statistics department

Find the number of pairs that include the Statistics (S) department representative. From the list, you can see that 4 pairs contain 'S'. Hence the probability of drawing a Statistics representative is 4 out of 10, or 0.4.

Computing the Probability That Committee Members Come From Laboratory Science Departments

Here, the Laboratory Science departments are considered to be Biology (B) and Chemistry (C). Hence we're looking for pairs that includes either B or C, or both. The combinations which satisfy this are: 'BC', 'BM', 'BP', 'BS', 'CM', 'CP'. Therefore, the probability that both committee members come from laboratory science departments is 6/10 or 0.6.