A 1. Blackjack Table
A standard deck of playing cards contains 52 cards divided equally into four suits: Spades, Hearts, Diamonds, and Clubs. Each suit contains thirteen cards: 2, 3, 4, 5, 6, 7, 8, 9, 10, J(ack), Q(ueen), K(ing), A(ce).
You walk up to the blackjack table at your local casino and ask to be dealt in. You are dealt two cards, the 5 of hearts and the 7 of spades. You see that the dealer also has two cards. One is face down, and the other is the 6 of clubs. You decide to let it ride and hope the dealer busts. The dealer's hidden card is revealed to be the 8 of clubs, making the dealer's hand worth 14 points. What is the probability that the dealer busts? You may assume that the dealer is usign only 1 deck, and that you have not observed the values of any other cards.
In blackjack, the worth of the dealer's hand is determined by summing the values of his individual cards. Numbered cards are worth their face value. Jacks, Queens, and Kings are worth 10 points, and Aces are worth either 11 or 1 point (in this question, Aces will always be worth 1 point). The dealer starts with 2 cards and repeats the following strategy:
If the dealer's hand is worth 16 or fewer points he draws another card. The value of the dealer's hand is now the value of his two starting cards, and all the cards that he has subsequently drawn.
When the dealer stops drawing, the value of his hand will be 17 or higher. If it is higher than 21, the dealer is said to have "bust" meaning that his hand is worthless. If it is between 17 and 21 (inclusive) then the dealer's hand is worth its value.
A2. Sadistic Tendencies
You are captured by a Russian spy and tortured for information. Every time you fail to answer his question adequately, he plays the following game:
First, he draws a number randomly between 1 and 4. This becomes the comparison value. He then draws a number from between 0 and 4. If it is greater than or equal to the comparison value, you get a lash with a wet noodle, and he draws again. He continues this until he draws a number that is less than the comparison value, at which point he continues his interrogation. If he draws a 3 as the comparison value, what is the expected number of lashes you will receive for that question?
What is the expected number of lashes you will receive per question?
A 3: University Admissions
The dean of a university is worried about gender discrimination. Despite having identical num-bers of applicants, the university, as a whole, seems to be admitting far more males than females.
However, when he investigated each department, he actually found that in every single department, the acceptance rate was slightly higher for women than for men.
What is the overall acceptance rate for men? For women? Try to explain the seeming contradiction between the dean's concerns and the department-level evidence.