Question

what is the theoretical probability of being dealt exactly two 2's in a 5-card hand from a standard 52-card deck?

Answer 1

The theoretical probability of being dealt exactly two 2's in a 5-card hand from a standard 52-card deck is:

`= (2162)/(32487)\\\\\approx 0.06655 \text{ in decimnal}`

Explanation:

The problem can be solved by using the combinatorics formula. The number of ways of drawing a subset of r items from a population of n items is given by

`^nC_r = (n!)/(r! (n-r)!)`

where n! is the factorial of n, r! the factorial of r and (n-r)! = factorial of (n-r)

The general formula for k! = k x (k - 1) x (k - 2) x ..... x 3 x 2 x 1

The number or ways in which you can get two 2's in a deal of 5 cards is given by

`^5C_2 = (5!)/( 2! (5 - 2)! ) \\\\= = (5!)/(2! * 3! )\\\\= 10`

Once we have been dealt 2 2's we have to compute how many ways we can get the remaining 3 cards. Since we are looking for exactly two 2's we cannot draw another 2

The number of cards left that we can draw the remaining three cards = 52(total cards) -2(two 2's already drawn) - 2(two 2's that cannot be drawn)

= 48 cards

We can draw 3 cards from 48 cards in
`^(48)C_3` ways

`^(48)C_3 = (48!)/( 3! (48 - 3)! )\\\\\\= (48!)/(3! * 45! )\\\\= 17296`

Therefore the total number of ways of drawing exactly two 2's
= 10 x 17296 = 172960

The number of ways in which we can draw 5 cards from 52 cards is given by

`^(52)C_5 = = (52!)/(5! (52 - 5)! )\\\\= (52!)/(5! * 47! )\\\\= 2598960`

P(exactly two 2's in a 5-card hand)

`= (172960)/(2598960)\\\\ \\= (2162)/(32487)\\\\`

or, in decimal

`\approx 0.06655`