Question

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards, find the probability that all the five cards are spades.

Answer 1

Let X = number of spade cards.

p = probability of drawing a spade card from pack of 52 cards.

Since, there are 13 spade cards in the pack of 52 cards,

∴ p = `13/52 = 1/4  and  "q" = 1 - "p" = 1 - 1/4 = 3/4`

Given: n = 5

∴ X ~ B`(5, 1/4)`

The p.m.f. of X is given by

P(X = x) = `"^nC_x  p^x  q^(n - x)`

i.e. p(x) = `"^5C_x  (1/4)^x  (3/5)^(5 - x)`, x = 0, 1, 2,...,5

P(all five cards are spade)

= P(X = 5) = p(5) = `"^5C_5(1/4)^5(3/4)^(5 - 5)`

`= 1(1/4)^5(3/4)^0`

`= 1 xx 1/1024 xx 1 = 1/1024`

Hence, the probability of all the five cards are spades = `1/1024`

Answer 2

Let X denote the number of spades.

P(getting spade) = p = `(13)/(52) = (1)/(4)`

∴ q= 1 – p = `1 - (1)/(4) = (3)/(4)`

Given, n = 5
∴  X ~ B`(5, 1/4)`
The p.m.f. of X is given by

P(X = x) = `""^5"C"_x(1/4)^x (3/4)^(5 - x), x` = 0, 1, ...,5

P(all five cards are spades)
= P(X = 5) 

= `""^5"C"_5(1/4)^5(3/4)^0`

= `(1)/4^5`.