# compute the probability that a random five-card hand has the following: 1.at least three cards with the same rank 2.at least two cards with the same rank

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Expert Answers

justaguide | Certified Educator

A deck of cards has 52 cards with 13 sets with 4 cards each that have the same rank.

The number of ways in which 5 cards can be dealt from a pack is `52C5 = 2598960`

The number of ways these 5 cards can have *at least* 3 cards with the same rank is `4C4*48C1*13 + 4C3*48C2*13 = 59280`

This gives the probability of at least 3 cards with the same rank as `59280/2598960 = 19/833`

The number of ways of getting at least two cards with the same rank is `4C2*48C3*13 + 4C3*48C2*13 + 4C4*48C1*13 = 1408368`

This gives the probability of *at least* 2 cards with the same rank as `1408368/2598960 = 2257/4165`

**The probability of at least 3 cards with the same rank is `19/833` and the probability of at least 2 cards with the same rank is `2257/4165` **