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Combinations - Problem 4

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Choosing in probability come in a lot when we're dealing with card games, so what we're looking at now is a probability of being dealt a heart flush in poker. And for those of you who don't play poker, what a heart flush means is you're dealt all hearts in a poker hand which is a 5 card hand.

So we are dealing with probability which tells us we have to do the ratio of the number of events that we want over the total number of potential events, we're dealing with a fraction. And the total number of events are just any possible poker hand that we can be dealt. There are 52 cards in a deck and so basically what we're dealing with is 52 choose 5, we're choosing 5 of them. The order we are dealt doesn't matter, so therefore I know it's a choose 52 cards in a deck choose 5 for your hand. So that's your total number of outcomes.

What we're then concerned with is the ways of being dealt a heart flush. Heart flush tells us all of our cards are hearts, so we know that we need to choose 5 to make our hand and what we're choosing 5 out of is only the hearts. So there are 52 cards in a deck 4 sits so 52 divided by 4 is the number of hearts which turns out ot be 13. So there are 13 choose 5 ways of getting only heart cards and there are 52 choose 5 different ways of being dealt a different hand and whatever this ratio is is going to be the probability of getting a hand completely filled with hearts.

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