Question

2. We are interested in analyzing data related to the Olympics from one decade. We are looking at individuals and if they participated in the summer or winter Olympics and whether or not they won a medal. Use S to denote summer and M to denote if a medal was won. The probability that someone participated in the summer Olympics is 72%. The probability that they won a medal is 13%. The probability that they won a medal and it was in the summer Olympics is 10%. ( please show steps)

a. What percentage of people participated in the summer Olympics or won a medal?

b. What percentage of people participated in the winter Olympics?

c. Given someone won a medal, what is the probability that they participated in the summer Olympics?

d. What percentage of people did NOT participate in the summer games NOR won a medal?

e. Are M and S mutually exclusive events? Why or why not? f. Are M and S independent events? Explain, using probabilities.

g. If we know someone participated in the summer Olympics, what is the probability that they also won a medal?

Answer 1

Solution Back-up Theory Percentage represents probability …………………………………………………………… (1) For 2 events, A and B, P(A ∪ B) = P(A) + P(B) - P(A ∩ B), i...................... ……………………………(2) If A and B are mutually exclusive, P(A ∩ B) = 0 ……………………………….(3) P(A ∪ B) is the same as P(at least one event of A and B) = 1 – P(neither A nor B) and so, P(neither A nor B) = 1 - P(A ∪ B)………….. .(4) P(AC) = 1 - P(A) ……………………………………......……………………………(5) If A and B are independent, P(A ∩ B) = P(A) x P(B) ....…………………………(6) If A and B are two events such that probability of B is influenced by occurrence or otherwise of A, then Conditional Probability of B given A, denoted by P(B/A) = P(B ∩ A)/P(A)..….(7) Now to work out the solution, As already given in the question, S represents the event that an individual participated in the summer Olympics. … (8) M represents the event that an individual won a medal ………...……………………(9) (8) and (9) => SC represents the event that an individual participated in the winter Olympics. … (8a) MC represents the event that an individual did not win a medal ……………………(9a) Given data then => P(S) = 0.72, P(M) = 0.13 and P(S ∩ M) = 0.1 ...……………… (10) Part (a) Percentage of people who participated in the summer Olympics or won a medal = P(S ∪ M) = 0.72 + 0.13 – 0.1 [vide (2) and (10)]= 0.75 ANSWER Part (b) Percentage of people who participated in the winter Olympics = P(SC) [vide (8a)] = 1 – 0.72 [vide (5) and (10)] = 0.28 ANSWER Part (c) Given someone won a medal, the probability that they participated in the summer Olympics = P(S/M) = P(S ∩M)/P(M) [vide (7)] = 0.1/0. 13[vide (10)] = 0.77 ANSWER Part (d) Percentage of people who did NOT participate in the summer games NOR won a medal = P(SC ∩ MC) = 1 - P(S ∪ M) [vide (4)] = 1 – 0.75 [vide Answer (a)] = 0.25 ANSWER Part (e) M and S are NOT mutually exclusive events. [vide (5) and (10)] ANSWER Part (f) M and S are NOT independent events. [vide (6) and (10); 0.1 ≠ 0.72 x 0.13] ANSWER Part (g) If we know someone participated in the summer Olympics, the probability that they also won a medal = P(M/S) = P(S ∩ M)/P(S) [vide (7)] = 0.1/0.72[vide (10)] = 0.14 ANSWER DONE