There are 52 cards in a standard deck. We want to randomly draw 5 cards.
1.) Use the formula for combinations to find how many ways to draw 5 cards from a standard deck.
C(n, x) = n! / x!*(n - x)! , where n is the sample size and x is the number of items selected.
C(52, 5) = 52! / [5!*(52 - 5)!] = 52! / (5!*47!) = 2,598,960
2.) There are 13 spades in a standard deck. We want to find how many ways to draw exactly 3 spades where order doesn't matter.
C(13, 3) = 13! / [3!*(13 - 3)!] = 13! / (3!*10!) = 286
3.) There are 2 slots left to fill. We have 39 remaining cards that aren't spades (clubs, diamonds, and hearts). We want to find how many ways to draw 2 cards that aren't spades where order doesn't matter.
C(39, 2) = 39! / [2!*(39 - 2)!] = 39! / (2!*37!) = 741
4.) P(draw exactly 3 spades) = [C(13, 3)*C(39, 2)] / C(52, 5) = 286*741 / 2,598,960 = 0.0815
The probability of randomly drawing 5 cards from a deck and getting exactly 3 spades is 0.0815.
