DRV Probability

Question

Suppose we have a standard deck of 52 cards (not including jokers). We are playing 5-card poker where players are handed 5 cards each. Find the average number of face cards we should expect in our hand.

Answer 1

since you are drawing 5 cards, the expected number of face cards is the number of cards drawn multiplied by the probability of any single card being a face card.

for this scenario that means taking

E(Face Cards) = 5 * P(Face Card) = 5 * 12/52 = 1.154

meaning that on average you can expect to have about 1.15 face cards in a 5-card poker hand

Answer 2

1. Find the probability of being dealt a single face card. 12 face cards out of 52 = 23% (p = 0.23)
2. Now think in terms of "you either receive a face card or not". Use the Binomial Probability Formula to find the probability of being dealt exactly 0 (x = 0) out of 5 (n = 5) face cards, 1 out of 5 face cards, 2 out of 5 face cards, and so on.

P(x) = (n! / x!(n - x)!) * p^x * (1 - p)^n-x

For example, the probability of receiving exactly 0 out of 5 cards is 27.1%

P(0) = (5! / 0!(5 - 0)!) * 0.23⁰ * (1 - 0.23)^5-0 = 0.271 or 27.1%

1. Once you repeat this formula 6 times for each possible outcome, sum the products of each outcome with its associated probability.

(0 * 0.271) + (1 * 0.404) + (2 * 0.242) + (3 * 0.072) + (4 * 0.011) + (5 * 0.001) = 1.153

The average number of face cards we should expect to see in our hand is 1.