91Ó°ÊÓ

Write a formula for the probability that a bridge hand (which is 13 cards chosen from an ordinary deck) has four aces, given that it has (at least) one ace. Write a formula for the probability that a bridge hand has four aces, given that it has the ace of spades. Which of these probabilities is larger?

A nickel, two dimes, and three quarters are in a cup. You draw three coins, one at a time, without replacement. What is the probability that the first coin is a nickel? What is the probability that the second coin is a nickel? What is the probability that the third coin is a nickel?

Given a random variable \(X\), how does the variance of \(c X\) relate to that of \(X\) ?

Evaluate the sum $$ \sum_{i=0}^{10} i\left(\begin{array}{c} 10 \\ i \end{array}\right)(.9)^{i}(.1)^{10-i} $$ which arose in computing the expected number of right answers a person would have on a 10 -question test with probability \(.9\) of answering each question correctly. First, use the binomial theorem and calculus to show that $$ 10(.1+x)^{9}=\sum_{i=0}^{10} i\left(\begin{array}{c} 10 \\ i \end{array}\right)(.1)^{10-i} x^{i-1} $$ Substituting \(x=.9\) almost gives the sum you want on the right side of the equation, except that in every term of the sum, the power on 9 is one too small. Use some simple algebra to fix this and then explain why the expected number of right answers is 9 .