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

Answer:

$\frac{11}{850}$

Step-by-step 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:

$\frac{13}{52} \cdot \frac{12}{51} \cdot \frac{11}{50} = \frac{1}{4} \cdot \frac{4}{17} \cdot \frac{11}{50} = \frac{1}{17} \cdot \frac{11}{50} = \frac{11}{850}$

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

$\binom{13}{3} = \frac{13!}{3!\cdot10!}$ ways.

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

$\binom{52}{3} = \frac{52!}{3!\cdot 49!}$

So the probability is:

$\frac{ \binom{13}{3} }{ \binom{52}{3} } = \frac{ \frac{13!}{3!\cdot 10!} }{ \frac{52!}{3!\cdot 49!} } = \frac{ \frac{13!}{10!} }{ \frac{52!}{49!} } = \frac{13 \cdot 12 \cdot 11}{52 \cdot 51 \cdot 50}$

Which again brings us to the result computed before.