Question

(52 cards) Five cards are dealt from a shuffled deck. What is the probability that the 5 cards contain: 1) exactly one ace 2) at least one ace

Answer 1

Answer:-1) The solution can be found as-

Number of ways of choosing one ace from 4 aces is = ⁴C₁ = 4
Number of ways of choosing 4 other cards from 48 (non ace cards) = ⁴⁸C₄ = 194580

Now, total number of ways of choosing 5 cards from 52 cards = ⁵²C₅ = 2598960

Thus probability of exactly one ace = (4 x 194580)/(2598960) = 0.2995 or 29.95 %

Answer:-2) The number of ways of choosing 5 cards with no ace = ⁴⁸C₅ = 1712304

Total number of ways of choosing 5 cards from 52 = ⁵²C₅ = 2598960

Thus probability of no ace = 1712304/2598960 = 0.6588

So probability of getting at least one ace = 1 - 0.6588 = 0.3412 or 34.12 %