Question

# Alice, Bob, and Charlie each pick random numbers from the set {1, 2, 3, 4, 5,...

Alice, Bob, and Charlie each pick random numbers from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} independently.

(a) What is the probability that Alice and Bob pick the same number?

(b) What is the probability that Alice, Bob, and Charlie all pick even numbers?

(c) What is the probability Alice’s pick is at most 4 (including 4) OR Bob’s pick is at least 8?

(d) Given that Alice picks 8, what is the probability that Charlie’s number is strictly larger than both Alice and Bob’s?

(a) Here let say Alice picked a random number then Bob will pick thesamenumber.

P(Alice and Bob pick the same number) = 10/10 * 1/10 = 0.1

(b) P(Alice, Bob and charlie all pic even numbers) = 5/10 * 5/10 * 5/10 = 0.125

(c) P(Alice pick is at most 4 or bob's pick is at least 8) = P(Alice pick is at most 4) + P(Bob's pick is at least 8) - P(ALice pickat most 8 and bob pick at least 8) =

= 4/10 * 1 * 1 + 1 * 1 * 3/10 - 4/10 * 1 * 3/10 = 0.58

(d) Alice picks 8,

so Charlier's number would strictly be larger than both Alice and Bob's.

Here the combinations are (8,8,9), (8,9,10)

Probability of such event = 2/100 = 0.02