1. Probability warm-up: conditional probabilities and Bayes rule

(a) Give the definition of the conditional probability of a discrete random variable X given a discrete random variable Y .

(b) Consider a biased coin with probability 3/4 of landing on heads and 1/4 on tails. This coin is tossed three times. What is the probability that exactly two heads occur (out of the three tosses) given that the first outcome was a head?

(c) Give two equivalent expressions of P(X, Y ):

(i) as a function of P(X) and P(Y |X)

(ii) as a function of P(Y ) and P(X|Y )

(d) Prove Bayes theorem:

(e) A survey of certain Montreal students is done, where 55% of the surveyed students are affiliated with UdeM while the others are affiliated with McGill. A student is drawn randomly from this surveyed group.

i. What is the probability that the student is affiliated with McGill?

ii. Now let’s say that this student is bilingual, and you know that 80% of UdeM students are bilingual while 50% of McGill students are. Given this information, what is the probability that this student is affiliated with McGill ?