What odds should a person give in favor of the following events?

• A card chosen at random from a 52-card deck is an ace.
• Two heads will turn up when a coin is tossed twice.
• Boxcars (two sixes) will turn up when two dice are rolled.

You offer [math]3 : 1[/math] odds that your friend Smith will be elected mayor of your city. What probability are you assigning to the event that Smith wins?

In a horse race, the odds that Romance will win are listed as [math]2 : 3[/math] and that Downhill will win are [math]1 : 2[/math]. What odds should be given for the event that either Romance or Downhill wins?

Let [math]X[/math] be a random variable with distribution function [math]m_X(x)[/math] defined by

• Let [math]Y[/math] be the random variable defined by the equation [math]Y = X + 3[/math]. Find the distribution function [math]m_Y(y)[/math] of [math]Y[/math].
• Let [math]Z[/math] be the random variable defined by the equation [math]Z = X^2[/math]. Find the distribution function [math]m_Z(z)[/math] of [math]Z[/math].

John and Mary are taking a mathematics course. The course has only three grades: A, B, and C. The probability that John gets a B is .3. The probability that Mary gets a B is .4. The probability that neither gets an A but at least one gets a B is .1. What is the probability that at least one gets a B but neither gets a C?

In a fierce battle, not less than 70 percent of the soldiers lost one eye, not less than 75 percent lost one ear, not less than 80 percent lost one hand, and not less than 85 percent lost one leg. What is the minimal possible percentage of those who simultaneously lost one ear, one eye, one hand, and one leg?^{[Notes 1]}

Notes

Assume that the probability of a “success” on a single experiment with [math]n[/math] outcomes is [math]1/n[/math]. Let [math]m[/math] be the number of experiments necessary to make it a favorable bet that at least one success will occur (see Exercise).

• Show that the probability that, in [math]m[/math] trials, there are no successes is [math](1 - 1/n)^m[/math].
• (de Moivre) Show that if [math]m = n \log 2[/math] then
  [[math]] \lim_{n \to \infty} \left(1 - \frac1n \right)^m = \frac12\ . [[/math]]
  Hint:
  [[math]] \lim_{n \to \infty} \left(1 - \frac1n \right)^n = e^{-1}\ . [[/math]]
  Hence for large [math]n[/math] we should choose [math]m[/math] to be about [math]n \log 2[/math].
• Would DeMoivre have been led to the correct answer for de Méré's two bets if he had used his approximation?

• For events [math]A_1, \dots, A_n[/math], prove that
  [[math]] P(A_1 \cup \cdots \cup A_n) \leq P(A_1) + \cdots + P(A_n)\ . [[/math]]
• For events [math]A[/math] and [math]B[/math], prove that
  [[math]] P(A \cap B) \geq P(A) + P(B) - 1. [[/math]]

If [math]A[/math], [math]B[/math], and [math]C[/math] are any three events, show that

Explain why it is not possible to define a uniform distribution function (see Definition) on a countably infinite sample space. Hint: Assume [math]m(\omega) = a[/math] for all [math]\omega[/math], where [math]0 \leq a \leq 1[/math]. Does [math]m(\omega)[/math] have all the properties of a distribution function?