Question

Cards are drawn from a well-shuffled deck of 52. Find the probability for each event below. a) Drawing a King or a heart. b)Drawing a Jack or a seven. c)Drawing a Jack and a seven.

Answer 1

We know that the probability is given by:

$P=\frac{\text{ number of favorable outcomes}}{\text{ total number of possible outcomes}}$

a)

Since a standard deck has 4 different suits this means that each one of them has 13 cards. Then we have 13 hearts, a standard deck has a total of 4 kings, since one of them is from the heart suit this means that in total we have 16 favorable outcomes. Hence the probability for this case is:

$P=\frac{16}{52}=\frac{4}{13}$

Therefore, the probability of drawing a king or a heart is 4/13

b)

In this case we have 4 possible outcomes for the jack and 4 possible outcomes for the seven, this means that in total we have 8 favorable outcomes, hence:

$P=\frac{8}{52}=\frac{2}{13}$

Therefore, the probability of drawing a jak or a seven is 2/13.

c)

In this case we have two different possibilities: A probability if we replace the first card drawn and a probability if we don't replace the first card drawn.

With replacement:

We know that we have 4 favorable outcomes for each drawn, to find the probability of getting both of the cards we want we need to multiply them, hence:

$P=\frac{4}{52}\cdot\frac{4}{52}=\frac{1}{13}\cdot\frac{1}{13}=\frac{1}{169}$

Therefore, the probability of getting a Jack and a seven with replacement is 1/169

Without replacent:

In this case we have a probability of 4/52 to get a jack (or a seven) in the first draw; for the second draw we have one card less in total in the deck this means that we have a probability of 4/51 of getting a seven (or a jack if the first one was seven). Hence the probability is:

$P=\frac{4}{52}\cdot\frac{4}{51}=\frac{1}{13}\cdot\frac{4}{51}=\frac{4}{663}$

Therefore, the probability of getting a jack and a seven without replacement is 4/663