Homework 17: Expectation

1. An urn contains four chips numbered 1 through 4. Two are drawn without replacement. Let the random variable \(X\) denote the larger of the two. Find \(E(X)\).
2. A fair coin is tossed three times. Let the random variable \(X\) denote the total number of heads that appear times the number of heads that appear on the first and third tosses. Find \(E(X)\).
3. A box contains 5 red and 5 blue marbles. Two marbles are drawn randomly. If they are the same color, then you win $1.10; if they are different colors, then you lose $1.00. Calculate the expected value your winnings.
4. An arrow is fired at random into a circle of radius 8. (Note that the probability that it lands in any region of the circle depends on the area of the region.) If its distance from the center is \begin{align} 0 \text{ to } 1 \text{ inches}:& \text{ win }$10\\ 1 \text{ to } 3 \text{ inches}:& \text{ win }$5\\ 3 \text{ to } 5 \text{ inches}:& \text{ win }$2\\ 5 \text{ to } 8 \text{ inches}:& \text{ lose }$4 \end{align} Find the expected winnings.
5. Suppose that teams A and B play in the world series, and that team A has a 60% chance of winning any game against team B. Find the expected number of games that will be played. (The series is the best of seven - the first team to win four games wins the series.)
6. Let \(X\) have pdf $$ f_X(x)=2(1-x), 0\leq x\leq 1 $$ Find \(E(X)\).