There are two decks of cards. One is complete, but the other is missing the ace
ID: 3044009 • Letter: T
Question
There are two decks of cards. One is complete, but the other is missing the ace of spades (A). Alice picks one of the two decks with equal probability and then selects a card from that deck uniformly at random. Write down the sample space and event of interest in addition to the probability for each:
1. What is the probability that Alice picked the complete deck, given that she selected the queen of diamonds?
2. What is the probability that Alice picked the complete deck, given that she selected a queen?
3. What is the probability that Alice picked the complete deck, given that she selected the ace of diamonds (A)?
4. What is the probability that Alice picked the complete deck, given that she selected an ace?
Explanation / Answer
1) probability of queen of diamonds =P(complete deck and queen of diamond+incomplete deck and queen of diamond)
=(1/2)*(1/52)+(1/2)*(1/51) =(!/2)*(103/(51*52))
therefore probability that Alice picked the complete deck, given that she selected the queen of diamonds
=P(complete deck and queen of diamond)/P(queen of diamond)=(1/2)*(1/52)/(!/2)*(103/(51*52))
=51/103
2)
P(queen) =P(complete deck and queen+incomplete deck and queen)=(1/2)*(4/52)+(1/2)*(4/51)
therefore probability that Alice picked the complete deck, given that she selected the queen
=P(complete deck and queen)/P(queen)=(1/2)*(4/52)/((1/2)*(4/52)+(1/2)*(4/51)) =51/103
3)simialry:
probability that Alice picked the complete deck, given that she selected the ace of diamonds (A)
=(1/2)*(1/52)/(!/2)*(103/(51*52)) =51/103
4)
P(ace) =P(complete deck and ace+incomplete deck and ace) =(1/2)*(4/52)+(1/2)*(3/51)
probability that Alice picked the complete deck, given that she selected an ace
=P(complete deck and ace)/P(Ace) =(1/2)*(4/52)/((1/2)*(4/52)+(1/2)*(3/51)) =(4*51)/(4*51+3*52)
=204/360 =17/30
(please revert for any clarifcation)