Question

A newly built casino is introducing a new gamble. Since this game is extremely new, the casino is offering a free play to everyone (no money or chips needed to gamble) so that all players get a sense of this new game. The game is played with the following rules

There are 3 decks of 20 cards each on the table:
• Deck A contains 20 red cards numbered 1–20.
• Deck B contains 10 red cards numbered 21–30 and 10 blue cards.
• Deck C contains 5 red cards numbered 31–35 and 15 blue cards.
Each of the 3 decks is shuffled, and 1 card is drawn from each deck. These 3 cards are shuffled and put
face down on the table, making a new pile of 3 cards. Let R be the number of red cards among these 3
cards.
a. Compute the expected value and the variance of R.

Parts b–d describe three different ways in which you could learn that the pile of 3 cards formed above
(with 1 card from each of the 3 decks) has 2 red cards. In each case, determine the probability that
the third card in the pile is also red. Note that these three parts are all independent—for example, the
information given in part b does not carry over to parts c or d.

b. The 3 cards in the new pile are turned over one at a time. The first card is the red 32, and the
second card is the red 5. What is the probability that the third card in the pile is also red?
c. The 3 cards in the new pile are turned over one at a time, but you only see the color on each card
(not the number). The first two cards flipped over are red. What is the probability that the third
card in the pile is also red?
d. You ask a friend to look at the 3 cards in the pile without showing you the cards. You ask them,
“Are there at least 2 red cards in the pile?” They confirm that yes, there are at least 2 reds in the
pile. What is the probability that all 3 cards are red?

Answer 1