Lec 2: A Gentle Introduction Flashcards
What makes a bet fair?
A bet is fair if the expected value is zero. So, the bet is fair is the long term average of gains/losses is zero
What is the equation for expected value? Why is it true?
$1 * (3/8) = 3/8th of a dollar = $0.375
What is an event tree? When do we use it?
What is an event?
An event is a set of outcomes
What are the events in the event tree? What is the probability of any one event?
1/8
How do you calculate probabilities?
What is the cardinality of the set? How do you calculate it?
[WebWork 1.16] In a toss game, you toss a certain number of dice and get a dollar if the outcomes are equal, zero otherwise. For example, if you toss two dice, you will get one dollar if the outcome is {1, 1}, or {2, 2}, or {3, 3}, or {4, 4}, or {5, 5}, or {6, 6}. Otherwise, you will get zero reward. Now, you toss 3 fair six-sided dice. What is the fair value of a ticket to play this game? A fair value of a ticket is the mean reward from playing this game.
6/(6*6*6)
[WebWork 1.17] You toss two 6-sided fair dice; one is green and the other red. You get a dollar if the red die outcome is larger than the green die outcome by exactly 5, and get zero dollars otherwise. What is the fair value of a ticket to play? A fair value of a ticket is the mean reward from playing this game?
1/(6*6)
[WebWork 1.18] You toss two identical 6-sided dies, you get a dollar if the difference between the outcomes is 3, zero otherwise. What is the fair value of a ticket to play this game? The fair value of a ticket is the mean reward from playing this game.
6/36
[WebWork 1.19] You toss a coin 3 times. You get a dollar if there is exactly one head, zero otherwise. For example, if you toss two coins, you will get one reward if the outcome is {H, T} or {T, H}, otherwise you get zero reward. What is the fair value of a ticket to play this game? The fair value of a ticket is the mean of the reward from playing this game.
3/8
[WebWork 1.20] You choose one out of 3 cards: 1 are red on one side and black on the other, one is red/red, and one is black/black. You place the chosen card on the table with a random side facing up. You will get a dollar if the opposite side of the chosen card has the same color as the face-up side, and zero otherwise. What is the fair value of a ticket to play this game? The fair value of a ticket is the mean reward from playing this game.
2/3
