Question

A hand of five cards is drawn from a standard 52 card deck.

(a) What is the probability of getting only one pair?

(b) What is the probability of getting all cards with the same suit?

Answer 1

A deck of 52 cards has 4 suits, Spades, Clubs, Diamonds and Hearts

There are 13 cards in each suit, from 2 till 10, the 3 face cards J, Q, K and finally Ace.

Therefore there are 4 cards of each type i.e 4 kings, 4 queens etc

Probability = Favourable outcomes/Total Outcomes

Please note ⁿC_x = n! / [(n-x)!*x!]

Total outcomes = ⁵²C₅ = 2598960

(a) Favourable outcomes:

Choose one of the 13 cards which will be the pair = ¹³C₁ = 13 possibilities (for eg we choose a King)

There will be 4 different cards ( 4 suits) of which we need to choose 2 = ⁴C₂ = 6 ways

Now we have 3 cards remaining to complete the set. We choose 3 different cards from the remaining 12 cards (as we cannot use the king) in ¹²C₃ = 220

Each of these 3 cards that we have chosen will have 4 of each type (4 suits) of which we need to choose 1 for each = ⁴C₁ * ⁴C₁⁴C₁ = 4 * 4 * 4 = 64

Therefore the total favourable outcomes = 13 * 6 * 220 * 64 = 1098240

Therefore the required probability = 1098240/2598960 = 0.4226

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(b) Favourable outcomes:

First we choose 1 suit out of 4 in ⁴C₁ = 4

Then we choose 5 cards out of 13 in ¹³C₅ = 1287

Therefore total favourable outcomes = 1287 * 4 = 5148

Therefore the required probability = 5148/2598960 = 0.00198 0.002

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