Pick a card statistics

Question

A card is chosen randomly from a standard deck. What is the probability it is an ace OR a red card? What is the probability is is an ace AND a red card?

Answer 1

A standard deck has 4 suits of 13 cards each, hearts and diamonds (both red) as well as spades and clubs (both black) so, consequently there are 26 red cards. Of course 2 of those red cards are aces, the aces of hearts and of diamonds. But there are two additional aces, the two black aces. That means we are trying to find one of 28 cards - 26 red cards and two black aces. This makes the probability 28/52 or, reduced, 7/13 or 53.8462%

For the second question, there are only 2 cards that are both aces and red making the probability 2/52 or , reduced, 1/26 or as a percent 3.8462%

You can also use the standard equations that pertain to "or" and "and" conditions:

Problem 1:

P(A or B) = P(A) + P(B) - P(A nd B) where A is "ace" and B is "red card"

P(A) = 4/52

P(B) = 26/52

P(A and B) = 2/52

So P(A or B) = 4/52 + 26/52 - 2/52 = 28/52

Problem 2:

P(A and B) = P(A)•P(B|A) where P(A) is the probability of aces and P(B|A) is the probability of a red card GIVEN that you have aces.

P(A) = 4/52

P(B|A) = 2/4

P(A and B) = 4/52 • 2/4 = 8/208 = 1/26