Question

A single card is drawn from a standard deck of playing cards. What is the probability that the card is a face card provided that a queen is drawn from the deck of cards?

(a) \(\frac{3}{13}\)

(b) \(\frac{1}{3}\)

(c) \(\frac{4}{13}\)

(d) \(\frac{1}{52}\)

I have been asked this question by my college professor while I was bunking the class.

I would like to ask this question from Discrete Probability in portion Discrete Probability of Discrete Mathematics

Answer 1

Right answer is (b) \(\frac{1}{3}\)

Best explanation: The probability that the card drawn is a queen = \(\frac{4}{52}\), since there are 4 queens in a standard deck of 52 cards. If the event is “this card is a queen” the prior probability P(queen) = \(\frac{4}{52} = \frac{1}{13}\). The posterior probability P(queen|face) can be calculated using Bayes theorem: P(king|face) = P(face|king)/P(face)*P(king). Since every queen is also a face card, P(face|queen) = 1. The probability of a face card is P(face) = (\(\frac{3}{13}\)). [since there are 3 face cards in each suit (Jack, Queen, King)]. Using Bayes theorem gives P(queen|face) = \(\frac{13}{3}*\frac{1}{13} = \frac{1}{3}\).