1. Suppose we have a six-sided die that we roll once. Let ai represent the event that the result is i. Let Bj represent the event that the dice comes out greater than j. Let C represent the event that the result is an even number.

  a) Since the number is greater than 1, what is the conditional probability that 3 is obtained? (P[A3|B1])

  b) Since the number is greater than 3, what is the conditional probability that 6 is obtained?

  c) Since the result is an even number, what is the conditional probability of a number greater than 3? (ie P[B3|C])

  d) Since the number is greater than 3, what is the conditional probability of an even number?

If you roll a die, what is the probability of rolling a number greater than 4?

4. A computer randomly prints a three-digit string of numbers. The string consists only of the digits 0, 1, 2, or 3 and satisfies the extra condition that the sum of its digits is 3. As an example, 021 is one such string. d. After the string of numbers is printed, we randomly select one of the strings three angs Each of the three digits is equally to be chosen. Let G be the event that the digit selected is a 0. This depends on the number of o's in the string that is printed. For example, if the string has two o's, we get the conditional probability P(G|A) = . Using similar logic, write down the conditional probabilities P(GB) and P(GC).

initial estimates of the probabilities of events are known as

a. joint probabilities

b. posterior probabilities

c. conditional probabilities

d.  prior probabilities