How many 5 cards hand dealt from a standard deck of playing cards contain exactly 3 face cards
|
I’m assuming that you mean that the five cards can be divided into three face cards, a king and a spade (and not just that they contain three cards, contain a king and contain a spade, possibly overlapping). Let’s count separately depending on whether the spade is also a face card. If the spade is a face card, we have $5$ face cards, of which at least one is a spade and at least one is a king. There are $\binom{12}5$ hands with $5$ face cards, of which $\binom95$ have no spade, $\binom85$ have no king and $\binom65$ have neither, so by inclusion–exclusion that makes $\binom{12}5-\binom95-\binom85+\binom65$. But now we’ve counted hands with only the king of spades and no other king and no other spade, of which there are $\binom64$, so we have to subtract those. If the spade is not a face card, we have one of $10$ non-face-card spades, and $4$ face cards, of which one is a king. There are $\binom{12}4$ sets of $4$ face cards, of which $\binom84$ have no king, so that makes $10\left(\binom{12}4-\binom84\right)$. Thus, in total, there are $\binom{12}5-\binom95-\binom85+\binom65-\binom64+10\left(\binom{12}4-\binom84\right)=4851$ such hands. Since this doesn’t agree with another answer that’s been posted, I wrote Java code to check it by enumeration. 1 Expert Answer
Jonah S. answered • 10/13/21 Math and Music Theory Tutor Specializing in AP, ACT, and SAT Test Prep Hi Ava! For this question, let's review some basic facts about a deck of cards. There are 52 cards in a deck, split evenly among 4 suits. This means each suit has 13 cards: 10 number cards (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10) and three face cards (Jack, Queen, King). This means that there are (3 face cards per suit)(4 suits per deck) = 12 face cards per deck. So we know that the probability of pulling one face card must be 12/52, and the probability of pulling one number card are 40/52. We know also that we're drawing a full hand, so we must not be replacing cards. In other words, our deck of cards will decrease in size by one card each time we draw. We should reflect this in a fraction: the probability of drawing a face card first is 12/52, but now that there are 11 face cards left and 51 cards left, the chance we draw a second face card is 11/51. We can repeat this for the other two face cards in our hand: the odds of pulling three face cards is (12/52)(11/51)(10/50). The method we use to figure out our odds of pulling a number card are similar: we want our last two cards to be number cards, of which there are still 40 left. But we have already pulled 3 cards from our deck, so the remaining cards total 49. Therefore, the probability of our fourth card being a number card is 40/49, and we symbolically remove a card from the top and bottom of that fraction to see that the probability of our fifth card being a number card are 39/48. Finally, we multiply all of our fractions together to represent the probability that these independent events happen at the same time. Our final answer for the probability of having 3 face cards and 2 number cards is (12/52)(11/51)(10/50)(40/49)(39/48) = 0.00660264, or 0.66203%. Still looking for help? Get the right answer, fast.ORFind an Online Tutor Now Choose an expert and meet online. No packages or subscriptions, pay only for the time you need. How many fiveSimilarly there are (13C3)x(39C1 )=286x39=11,154 hands containing exactly three clubs.
What is the probability of getting 3 of a kind in a 5 card hand?5-card poker hands. How many 5 cards hands dealt from a standard deck of cards contain exactly the same suit?Usually you just separate the cards you need from a deck of poker or pinochle cards. How many 5-card hands are there where all the cards have the same suit (consider 52 cards with 13 of each of the four suits)? There are = 1287 “flushes” in each of the four suits, so there are 1287 * 4 = 5148 total.
How many 5 card hands can be made if more face cards than non face cards?= 792, since there are 12 face cards and you want 5 of them. The probability of getting a 5 card hand made entirely of face cards is the above number divided by 52 nCr 5 since that is the total number of 5 card hands that can be made.
|
