Chapter 7: Discrete Probability

Question: Let\({X_n}\)be the random variable that equals the number of tails minus the number of heads when\(n\)fair coins are flipped.

1. What is the expected value of\({X_n}\)
2. What is the variance of\({X_n}\)

Question: Let \(X\) be a random variable on a sample space \(S\). Show that \(V(aX + b) = {a^2}V(X)\) whenever \(a\) and \(b\) are real numbers.

Question: What is the expected number of heads that come up when

a fair coin is flipped 10 times?

Question: Suppose that $\mathit{E}$and$\mathit{F}$are events in a sample space and$\mathit{p}\mathbf{\left(}\mathit{E}\mathbf{\right)}\mathbf{=}\frac{\mathbf{2}}{\mathbf{3}}\mathbf{,}\mathit{p}\mathbf{\left(}\mathit{F}\mathbf{\right)}\mathbf{=}\frac{\mathbf{3}}{\mathbf{4}}\mathbf{,}$and $\mathit{p}\mathbf{(}\mathit{F}\mathbf{\mid }\mathit{E}\mathbf{)}\mathbf{=}\frac{\mathbf{5}}{\mathbf{8}}$. Find$\mathit{p}\mathbf{(}\mathit{E}\mathbf{\mid }\mathit{F}\mathbf{)}$.

Question: What is the probability that a fair die comes up six when it is rolled?

Question: Find the probability of each outcome when a loaded die is rolled, if a 3 is twice as likely to appear as each of the other five numbers on the die.

Question:\(P({\rm{ tails }}) = \frac{1}{4},P({\rm{ heads }}) = \frac{3}{4}\)

a) What conditions should be met by the probabilities assigned to the outcomes from a finite sample space?

b) What probabilities should be assigned to the outcome of heads and the outcome of tails if heads come up three times as often as tails?

A player in the Mega Millions lottery picks five different integers between \(1\) and \(56\) , inclusive, and a sixth integer between \(1\) and \(46\) , which may duplicate one of the earlier five integers. The player wins the jackpot if the first five numbers picked match the first five numbers drawn and the sixth number matches the sixth number drawn.

a) What is the probability that a player wins the jackpot?

b) What is the probability that a player wins \(250,000\), which is the prize for matching the first five numbers, but not the sixth number, drawn?

c) What is the probability that a player wins \(150\) by matching exactly three of the first five numbers and the sixth number or by matching four of the first five numbers but not the sixth number?

d) What is the probability that a player wins a prize, if a prize is given when the player matches at least three of the first five numbers or the last number.

Question:What is the probability that a player of a lottery wins the prize offered for correctly choosing five (but not six)numbers out of six integers chosen at random from the integers between 1 and 40, inclusive?

Question: Show that if\(X\) and \(Y\) are independent random variables, then \(V\left( {XY} \right) = E{\left( X \right)^2}V\left( Y \right) + E{\left( Y \right)^2}V\left( X \right) + V\left( X \right)V\left( Y \right)\)