Simulate the Monty Hall problem. Carefully state any assumptions that you have made when writing the program. Which version of the problem do you think that you are simulating?
Two cards are drawn from a bridge deck. What is the probability that the second card drawn is red?
If [math]P(\tilde B) = 1/4[/math] and [math]P(A|B) = 1/2[/math], what is [math]P(A \cap B)[/math]?
- What is the probability that your bridge partner has exactly two aces, given that she has at least one ace?
- What is the probability that your bridge partner has exactly two aces, given that she has the ace of spades?
Prove that for any three events [math]A[/math], [math]B[/math], [math]C[/math], each having positive probability, and with the property that [math]P(A \cap B) \gt 0[/math],
Prove that if [math]A[/math] and [math]B[/math] are independent so are
- [math]A[/math] and [math]\tilde B[/math].
- [math]\tilde A[/math] and [math]\tilde B[/math].
A doctor assumes that a patient has one of three diseases [math]d_1[/math], [math]d_2[/math], or [math]d_3[/math]. Before any test, he assumes an equal probability for each disease. He carries out a test that will be positive with probability .8 if the patient has [math]d_1[/math], .6 if he has disease [math]d_2[/math], and .4 if he has disease [math]d_3[/math]. Given that the outcome of the test was positive, what probabilities should the doctor now assign to the three possible diseases?
In a poker hand, John has a very strong hand and bets 5 dollars. The probability that Mary has a better hand is .04. If Mary had a better hand she would raise with probability .9, but with a poorer hand she would only raise with probability .1. If Mary raises, what is the probability that she has a better hand than John does?
The Polya urn model for contagion is as follows: We start with an urn which contains one white ball and one black ball. At each second we choose a ball at random from the urn and replace this ball and add one more of the color chosen. Write a program to simulate this model, and see if you can make any predictions about the proportion of white balls in the urn after a large number of draws. Is there a tendency to have a large fraction of balls of the same color in the long run?