Question

Find the probability of being dealt a hand of 12 cards containing 3 ​spades, 5 ​hearts, 2 ​diamonds, and 2 clubs from an ordinary deck of 52 cards. The probability of being dealt a hand containing 3 ​spades, 5 ​hearts, 2 diamonds, and 2 clubs is

Answer 1

Solution

Back-up Theory

Probability of an event E, denoted by P(E) = n/N ……………………………………….…..………(1)

where n = n(E) = Number of outcomes/cases/possibilities favourable to the event E and

N = n(S) = Total number all possible outcomes/cases/possibilities.

Number of ways of selecting r things out of n things is given by ⁿC_r = (n!)/{(r!)(n - r)!}……….…(2)

Values of ⁿC_r can be directly obtained using Excel Function: Math & Trig COMBIN……....…. (2a)

Now, to work out the solution,

There are 13 cards each of spades, ​hearts, diamonds, and2 clubs. So, vide (2), number of selections of 3 ​spades, 5 ​hearts, 2 diamonds, and 2 clubs is: (¹³C₃) (¹³C₅)(¹³C₂)(¹³C₂)

= 284 x 1287 x 78 x 78 [vide (2a)]

So, vide (1), n = 2239410888

Now, 12 out of 52 cards can be selected, vide (2), in (⁵²C₁₂) ways

= 2.06379E+11 [vide (2a)]

So, vide (1), N = 2.06379E+11

Hence, vide (1),

The probability of being dealt a hand containing 3 ​spades, 5 ​hearts, 2 diamonds, and 2 clubs is:

2239410888/2.06379E+11

= 0.0109 Answer

DONE