What is the probability of getting 5 hearts when dealt 5 cards in a poker game?
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To find the probability, we need to find the fraction where the numerator is the number of ways to have a flush and the denominator is the number of 5 card hands possible. Show
Each of these numbers will be found using Combinations (we don't care about the order of the draw, merely of what ends up in our hand). The general formula for Combinations is: #C_(n,k)=(n!)/((k)!(n-k)!)# with #n="population", k="picks"# First let's calculate the denominator (picking 5 random cards from a pack of 52 cards): #C_(52,5)=(52!)/((5)!(52-5)!)=(52!)/((5!)(47!))# Let's evaluate it! #(52xx51xxcancelcolor(orange)(50)^10xx49xxcancelcolor(red)48^2xxcancelcolor(brown)(47!))/(cancelcolor(orange)5xxcancelcolor(red)(4xx3xx2)xxcancelcolor(brown)(47!))=52xx51xx10xx49xx2=2,598,960# And now let's calculate the numerator. We're going to calculate all hands that involve a flush (so not just a flush but also a straight flush and royal flush) so that we'll be looking at any hand that has five cards of the same suit (with a suit having 13 cards in total). We can express getting this with: #C_(13,5)# Keep in mind that there are 4 suits this can happen in, but we only want 1, and so we need to multiply by #C_(4,1)#. Putting it together then, we get: #C_(4,1)xxC_(13,5)=(4!)/((1!)(4-1)!)xx(13!)/((5!)(13-5)!)=(4!13!)/(3!5!8!)# Let's evaluate this! #(cancelcolor(red)(4!)xx13xx12xx11xxcancelcolor(brown)10xxcancelcolor(blue)9^3xxcancelcolor(orange)(8!))/(cancelcolor(blue)3xxcancelcolor(brown)(2xx5)xxcancelcolor(red)(4!)xxcancelcolor(orange)(8!))=13xx12xx11xx3=5148# (keep in mind that we've just calculated all hands that have a flush element to them, including straight flushes and royal flushes!) The probability of getting a hand with a flush is: #5148/2598960~=.00198# ~~~~~ To exclude straight and royal flushes (these are hands that have 5 consecutive value cards in the same suit, such as 3, 4, 5, 6, 7 of hearts), we can deduct those possibilities from the 5148 flush hands. There are 10 different ways to get a straight (A-5, 2-6, 3-7,..., 10-A) and 4 suits, so we can deduct #4xx10=40# hands, so that's #5148-40=5108# hands, giving: #5108/2598960~=.00196#
University of Michigan 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! Thank you for watching the video.To unlock all 5,300 videos, start your free trial. Carl Horowitz University of Michigan 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! Share
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.
(most recent edit: January 2, 2005)A SINGLE PAIRThis the hand with the pattern AABCD, where A, B, C and D are from the distinct "kinds" of cards: aces, twos, threes, tens, jacks, queens, and kings (there are 13 kinds, and four of each kind, in the standard 52 card deck). The number of such hands is (13-choose-1)*(4-choose-2)*(12-choose-3)*[(4-choose-1)]^3. If all hands are equally likely, the probability of a single pair is obtained by dividing by (52-choose-5). This probability is 0.422569.TWO PAIRThis hand has the pattern AABBC where A, B, and C are from distinct kinds. The number of such hands is (13-choose-2)(4-choose-2)(4-choose-2)(11-choose-1)(4-choose-1). After dividing by (52-choose-5), the probability is 0.047539.A TRIPLEThis hand has the pattern AAABC where A, B, and C are from distinct kinds. The number of such hands is (13-choose-1)(4-choose-3)(12-choose-2)[4-choose-1]^2. The probability is 0.021128.A FULL HOUSEThis hand has the pattern AAABB where A and B are from distinct kinds. The number of such hands is (13-choose-1)(4-choose-3)(12-choose-1)(4-choose-2). The probability is 0.001441.FOUR OF A KINDThis hand has the pattern AAAAB where A and B are from distinct kinds. The number of such hands is (13-choose-1)(4-choose-4)(12-choose-1)(4-choose-1). The probability is 0.000240.A STRAIGHTThis is five cards in a sequence (e.g., 4,5,6,7,8), with aces allowed to be either 1 or 13 (low or high) and with the cards allowed to be of the same suit (e.g., all hearts) or from some different suits. The number of such hands is 10*[4-choose-1]^5. The probability is 0.003940. IF YOU MEAN TO EXCLUDE STRAIGHT FLUSHES AND ROYAL FLUSHES (SEE BELOW), the number of such hands is 10*[4-choose-1]^5 - 36 - 4 = 10200, with probability 0.00392465A FLUSHHere all 5 cards are from the same suit (they may also be a straight). The number of such hands is (4-choose-1)* (13-choose-5). The probability is approximately 0.00198079. IF YOU MEAN TO EXCLUDE STRAIGHT FLUSHES, SUBTRACT 4*10 (SEE THE NEXT TYPE OF HAND): the number of hands would then be (4-choose-1)*(13-choose-5)-4*10, with probability approximately 0.0019654.A STRAIGHT FLUSHAll 5 cards are from the same suit and they form a straight (they may also be a royal flush). The number of such hands is 4*10, and the probability is 0.0000153908. IF YOU MEAN TO EXCLUDE ROYAL FLUSHES, SUBTRACT 4 (SEE THE NEXT TYPE OF HAND): the number of hands would then be 4*10-4 = 36, with probability approximately 0.0000138517.A ROYAL FLUSHThis consists of the ten, jack, queen, king, and ace of one suit. There are four such hands. The probability is 0.00000153908.NONE OF THE ABOVEWe have to choose 5 distinct kinds (13-choose-5) but exclude any straights (subtract 10). We can have any pattern of suits except the 4 patterns where all 5 cards have the same suit: 4^5-4. The total number of such hands is [(13-choose-5)-10]* (4^5-4). The probability is 0.501177.
What is the probability you are dealt 5 cards of the same suit?Here all 5 cards are from the same suit (they may also be a straight). The number of such hands is (4-choose-1)* (13-choose-5). The probability is approximately 0.00198079.
What is the probability of getting a full house in a 5 card poker hand?Thus the probability would be C(5,3)×52×3×2×48×352×51×50×49×48≈0.00144.
When 5 cards are dealt from a standard deck of 52 cards what is the probability of getting 4 aces?If you have a standard deck of 52 cards, what is the probability that out of a hand of 5 cards you get 4 aces? Then the # of hands which has 4 aces is 48 (because the 5th card can be any of 48 other cards). So there is 1 chance in (2,598,960/48) = 54,145 of being dealt 4 aces in a 5 card hand.
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