A card is drawn at random from a well-shuffled pack of 52 playing cards. If it is a face card (king, queen or jack), a ball is drawn at random from Bag I, which contains 3 red and 1 white ball; otherwise a ball is drawn at random from Bag II, which contains 1 red and 3 white balls. The probability that the ball drawn is red is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{19}{52}\)
A pack has 12 face cards and 40 other cards. Let \(E_1\) and \(E_2\) be the events that Bag I and Bag II are used, and R the event that the ball is red.
\( \displaystyle P(E_1) = \frac{12}{52},\quad P(E_2) = \frac{40}{52} \)
\( \displaystyle P(R|E_1) = \frac{3}{4},\quad P(R|E_2) = \frac{1}{4} \)
By the theorem of total probability:
\( \displaystyle P(R) = \frac{12}{52}\times\frac{3}{4}+\frac{40}{52}\times\frac{1}{4} \)
\( \displaystyle P(R) = \frac{9}{52}+\frac{10}{52} = \frac{19}{52} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{1}{2}\) — Incorrect. This treats the two bags as equally likely; Bag II is used much more often, since only 12 of the 52 cards are face cards.
Option (3) \(\displaystyle\frac{33}{52}\) — Incorrect. This is \(1-\frac{19}{52}\), the chance that the ball is white.
Option (4) \(\displaystyle\frac{3}{13}\) — Incorrect. This is \(\frac{12}{52}\), the chance that Bag I is used.