# Probability of Chosing a King Next

An ordinary deck of playing cards is shuffled and the top card set aside without knowing its face value. What is the probability that the next card is a king?

P(king)= $3/51$....I am stuck because I am assuming the first card I Put down is/not a king determines the chance the at my next card is a king. Either way I know my sample is 51 but were Am I going wrong?

• Here is a well-mixed deck of cards. What is the probability that the top card is a king? What is the probability that the bottom card is a king? What is the probability that the 17th card from the top is a king? What is the probability that the 2nd card from the top is a king? Do you really think kings are more likely to turn up is some positions that in others? – bof Apr 5 '17 at 2:50
• Let $p_1$ be the probability that the second card dealt is an ace, $p_2$ the probability that it's a deuce, $p_3$ the probability that it's a trey, and so on, so that $p_{13}$ is the probability that it's a king. Are we agreed that those $13$ probabilities add up to $1?$ Now, do you think the probabilities are all equal? Or do you think some of those events are more likely than others? – bof Apr 5 '17 at 2:52
• I've shuffled the deck and I'm about to deal a card. The probability that I'm going to deal a king is $1/13,$ right? But wait, the player next to me wants to cut the cards. He cuts, moving an unknown number of kings to the lower part of the deck. Now what's the probability of dealing a king?? – bof Apr 5 '17 at 3:02

## 2 Answers

Assuming the card is a king, and that has $\frac{4}{52}$ probability, then the probability that the next card is a king is $\frac{3}{51}$ as you correctly identified. But, if the card is not a king, and that has $\frac{48}{52}$ probability, then the probability the next card is a king is $\frac{4}{51}$. Hence the answer by the law of total probability is,

$$\frac{4}{52}(\frac{3}{51})+\frac{48}{52}(\frac{4}{51})$$

$$=\frac{4(3)+4(48)}{51(52)}$$

$$=\frac{4(3+48)}{51(52)}$$

$$=\frac{4}{52}$$

• ... which simplified is equal to $\frac{4}{52}$ – JMoravitz Apr 5 '17 at 3:00

It's 4/52, as you don't know the value of the first card.

Think of it this way--there are 52 cards in front of you and all you know is 4 of them are kings and 48 are not.

You can also calculate directly using conditional probability:

$P(King On Second Card)=P(King On Second Card|King On First Card)\cdot P(King On First Card) + P(King On Second Card|Not King On First Card)\cdot P(Not King On First Card)=\frac 3 {51}\cdot \frac 4 {52} + \frac 4 {51}\cdot \frac {48} {52}$