Compute the probability that a five-card poker hand is dealt to you that contains a royal flush? Please explain the combination or permutation formulas used. step by step thx
i think it is (4C1/52)(4C1/51)(4C1/50)(4C1/49)(4C1/48) solve and then divide by the total amount of options which is 52c5. i'm not sure if this should be a Permetation or a combination as I outlined above
Answers
While it is a tricky question, the answer to calculate total number of poker hands depends on the total number of players. Let’s assume that you are playing a game of five-card showdown with two players, thus there are a total of ten cards that are dealt.
We can calculate the total number of permutations to calculate how these ten cards can be dealt, subjected to the condition that the first card is a random card (out of the 52 cards in all) and all subsequent cards are taken from the remaining pack of cards one after another. We will also assume that it is an ideal case and mathematical rules can be applied without any exceptions.
P = (52×51×50×49×48×47×46×45×44×43) = 5.740770388953600e16
The result is a 16 digit number.
A more difficult situation can prevail if cards are dealt in a game of bridge. In such a case, the exact number of combinations is 52! That approximates to a total number of permutations as 8.0658175e+67 (or 80,658,175,170,943,878,571,660,636,856,403,766,976,000,000,000,000,000,000,000,000,000,000). It is a grand number indeed!
But here is a special case- we are not considering the order in which these five players have received their respective cards. As such, we should deduct that number from this total number above. The number for the exceptional condition is C = 5x4x3x2 = 120 combinations to denote those different sequences in which those five cards can be arranged in.
The result is P/(2C) = 239,198,766,206,400. This means the more the number of players, the higher is the number of possible games.
Alternatively, if you would like to know about the total number of combinations for an exactly 5 card hand, here’s the calculation:
(We are assuming there is only one player in this scenario).
P = 52x52x50x49x48 = 311,875,200 permutations
C = 5x4x3x2x1 = 120 combinations
So the result is P/C = 311,875,200/120 = 2,598,960 hands.
In short, you can use this formula [n!/[(n-k)!*k!] to calculate total number of card combinations in a random group of cards, where k is the random cards taken out of total number of n cards.
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