Question

2. A standard 52-card deck consists of 4 suits (hearts, diamonds, clubs, and spades). Each suit has 13 cards: 10 are pip cards (numbered 1, or ace, 2 through 10) and 3 are face cards (jack, queen, and king).

You randomly draw a card then place it back. If it is a pip card, you keep the deck as is. If it is a face card, you eliminate all the pip cards. Then, you draw a new card. What is the probability you draw the queen of hearts in the end?

Answer 1

total number of cards = 52

number of face cards = 12 ( each suit has 3 face cards )

number of pip cards = 40 ( each suit has 10 pip cards )

P[ Drawing a face card ] = 12/52

If a pip card is drawn in the first trial, you keep the deck as is.

P[ draw the queen of hearts in the end | a pip card is drawn in the first trial ] = 1/52 ( only 1 queen of hearts in one deck )

P[ Drawing a pip card ] = 40/52

If it is a face card, you eliminate all the pip cards.

Number of remaining cards = 12 ( all face cards )

P[ draw the queen of hearts in the end | a face card is drawn in the first trial ] = 1/12 ( only 1 queen of hearts in one deck )

P[ draw the queen of hearts in the end ] = P[ draw the queen of hearts in the end | a face card is drawn in the first trial ]*P[ Drawing a pip card ] + P[ draw the queen of hearts in the end | a pip card is drawn in the first trial ]*P[ Drawing a face card ]

P[ draw the queen of hearts in the end ] = (1/12)*(12/52) + (1/52)*(40/52)

P[ draw the queen of hearts in the end ] = (1/52) + (40/2704)

P[ draw the queen of hearts in the end ] = ( 52 + 40 )/2704

P[ draw the queen of hearts in the end ] = 92/2704

P[ draw the queen of hearts in the end ] = 0.034