Numbered Balls
Suppose you have a bag containing seven balls numbered 0,1,1,2,3,5,8.
(a) You perform the following experiment: pull out a single ball and record its number. What is the expected value of the number that you record?
(b) You repeat the experiment from part (a), except this time you pull out two balls together and record their total. What is the expected value of the total that you record?
2 How Many Queens?
You shuffle a standard 52-card deck, before drawing the first three cards from the top of the pile. Let X denote the number of queens you draw.
(a) What is P(X = 0)?
(b) What is P(X = 1)?
(c) What is P(X = 2)? (d) What is P(X = 3)?
(e) Do the answers you computed in parts (a) through (d) add up to 1, as expected?
(f) Compute E(X) from the definition of expectation.
(g) Suppose we define indicators Xi, 1 ≤ i ≤ 3, where Xi is the indicator variable that equals 1 if the ith card is a queen and 0 otherwise. Compute E(X) using linearity of expectation.
(h) Are the Xi indicators independent? Does this affect your solution to part (g)?
3 More Aces in a Deck
There are four aces in a deck. Suppose you shuffle the deck; define the random variables:
X1 = number of non-ace cards before the first ace
X2 = number of non-ace cards between the first and second ace
X3 = number of non-ace cards between the second and third ace
X4 = number of non-ace cards between the third and fourth ace X5 = number of non-ace cards after the fourth ace
1. What is X1 +X2 +X3 +X4 +X5?
2. Argue that the Xi random variables all have the same distribution. Are they independent?
3. Use the results of the previous parts to compute E(X1).
