Question

You draw a card uniformly at random from a standard deck, then remove all cards of strictly higher rank (e.g., if you draw a 4, then the remaining deck consists of four 2s, four 3s, and three 4s). You repeat this process three times on the same deck, without putting the removed cards back. What is the probability that the three cards you’ve drawn are a 3 and two 5s?

Answer 1

`(2)/(455)` ≈ 0.0044

Explanation:

The probability that the three cards you’ve drawn random from a standard deck are a 3 and two 5s is possible only when you

• first draw 5
• secondly draw 5
• thirdly draw 3

If one draws 3 first, then there is no probability that one draws 5 after, because all 5's are need to be removed after drawing 3.

The probability drawing 5 is
`(4)/(52)` . After removing cards with higher rank, remaining cards are 15. (four 2's, four 3's, four 4's and three 5's)

The probability of drawing another 5 is then
`(3)/(15)`. No card is removed else, since 5 is the highest rank. Thus, 14 cards left.

The probability of drawing 3 after two 5 is
`(4)/(14)`

To find all these events happening successively, we need the multiply their probabilities:

`(4)/(52)` ×
`(3)/(15)` ×
`(4)/(14)` =
`(2)/(455)` ≈ 0.0044