Question

Three cards are selected one after the other from a standard deck of 52 cards.

What is the probability that all three cards are spades if none of the first two cards is replaced?

Answer 1

`(11)/(850)`

Explanation:

Let's see

The probability of the first card being a spade is 13/52 (as there's 13 spades in the deck.

The probability for the second card is 12/51 (because there's 12 spades left in 51 cards total).

The probability for the third is - you guessed it! - 11/50

so the total probability is:

`(13)/(52) \cdot (12)/(51) \cdot (11)/(50) = (1)/(4) \cdot (4)/(17) \cdot (11)/(50) = (1)/(17) \cdot (11)/(50) = (11)/(850)`

A more generic solution:

Let's use the binomials to find the solution with combinations:

We need to pick 3 cards out of 13. This can be done in

`\binom{13}{3} = (13!)/(3!\cdot10!)` ways.

And the total number of ways to pick 3 cards out of 52 is:

`\binom{52}{3} = (52!)/(3!\cdot 49!)`

So the probability is:

`\frac{ \binom{13}{3} }{ \binom{52}{3} } = ( (13!)/(3!\cdot 10!) )/( (52!)/(3!\cdot 49!) ) = ( (13!)/(10!) )/( (52!)/(49!) ) = (13 \cdot 12 \cdot 11)/(52 \cdot 51 \cdot 50)`

Which again brings us to the result computed before.