Suppose you draw two cards from a deck of 52 cards without replacement.

Question

Suppose you draw two cards from a deck of 52 cards without replacement.

1) What is the probability that both cards are hearts?

2) What is the probability that exactly one of the cards is hearts?

3) What is the probability that none of the cards are hearts?

I get that the first answer is 13/52 * 12/51.

For the second, is it 13/52 * 39/51 because the new deck has 51 cards of which 12 are hearts, hence 39/51 is the probability of the second not being hearts?

Will the third answer just be 39/52 * 38/51?

Answer 1

You're missing something in part 2), namely that drawing a heart first and drawing a non-heart first are both allowed.

You should multiply your answer by 2 since there are 2 ways of drawing exactly 1 heart: heart first, or non-heart first.

Your other answers look fine.

Answer 2

You can see your question matches the following assumptions:

1. The result of each draw (the elements of the population being sampled) can be classified into one of two mutually exclusive categories (e.g. Hearts/No Hearts).
2. The probability of a success changes on each draw, as each draw decreases the population (sampling without replacement from a finite population).

This means you could be able to solve it via the hypergeometric distribution:

https://en.wikipedia.org/wiki/Hypergeometric_distribution

This gives you for

a) 0.058824% (same as your answer)

b) 0.382353% (your answer times 2)

c) 0.558824% (same as your answer)