Question

Percy shuffles a standard $52$-card deck and starts turning over cards one at a time, stopping as soon as the first spade is revealed. what is the expected number of cards that percy turns over before stopping (including the spade)? (note: there are $13$ spades in a deck.)

Answer 1

The expected number of cards that Percy turns over before stopping (including the spade) is approximately 6.5.

Step-by-step explanation:

To find the expected number of cards that Percy turns over before stopping (including the spade), we need to calculate the average number of cards turned over for each possible outcome. The probability of turning over each card is given by the ratio of the number of spades remaining in the deck to the number of cards remaining. Initially, there are 13 spades and 52 cards in total, so the probability of turning over a spade on the first card is 13/52. If Percy doesn't turn over a spade on the first card, there are now 12 spades remaining and 51 cards in total. So the probability of turning over a spade on the second card is 12/51. Continuing this pattern, the expected number of cards turned over before stopping can be calculated as:

Expected number of cards = (13/52) * 1 + (12/51) * 2 + (11/50) * 3 + ... + (1/40) * 13

Simplifying this expression gives the expected number of cards as approximately 6.5.

Answer 2

3 because it’s 1/4 so you would turn over 3 then the fourth one would be expected to be a spade