A card is drawn from a shuffled deck of 52 cards, and not replaced. Then a second card is drawn. What is the probability that the second card is a king?

Question

Elena Atkins · Accepted Answer

#1/13#

To give a more explained solution of this problem, there are 2 cases you have to consider:

Case 1: The first card drawn is a king Case 2: The first card drawn is not a king

The reason there's a difference is because in Case 1 the taking of a king on the first card means there is a smaller chance of getting a king on the second card (because the originally taken card is not replaced).

To get the probability of the 2nd card being a king, we can find each individual probability for Cases 1 and 2 and add them together since each of those possibilities are disjoint; in other words, it's not possible that the first card drawn is a king and not a king at the same time.

Case 1

If the first card drawn is a king, the probability of that happening is #4/52 = 1/13#. The probability of the 2nd card being a king as well would then be #3/51#, since there is one less king possible to be drawn. Multiplying these together gives us: #1/13 * 3/51 = 3/663 = 1/221#

Case 2

If the first card drawn is not a king, the probability of that happening is #48/52 = 12/13#. (This is because there are 48 cards we're interested in which aren't kings, out of 52 total cards). The 2nd card being a king has a probability of #4/51#, since all 4 kings are still available out of 51 cards left in the deck. Multiplying these together gives us: #12/13 * 4/51 = 48/663 = 16/221#.

Answer

Adding these 2 possibilities together gives the overall probability of drawing a king on the 2nd draw: #1/221 + 16/221 = 17/221 = 1/13#