**Exercise 1**

**a.**- If
*A*and*B*are independent,

If*A*and*B*are independent, then not*A*and*B*are independent. So,

Since and , then . **b.**- First note that

Since*A*and*B*are independent we get that

**c.**- Let

We have

Since (*A*wins the second match) we get that . Hence,

**Exercise 2**

**a.**- As usual, a good start is to write out the probability information stated in the problem.

We now that and . By the multiplication principle we find

Repeating this process for all combination of results and ratings we construct the table of joint probability.

Panel rating Poor Fair Good Failure 0.30 0.18 0.12 Result Moderate 0.06 0.12 0.12 Major 0.01 0.03 0.06

**b.**- The tree diagram is shown below:
For the first branch of a probability tree, we pick the events that have known unconditional probabilities. In this case, that's the introduction results - failure, moderate, or major. For the next branches, we use the ratings - poor, fair, or good - and insert conditional probabilities given the preceding branch. Multiplying the probabilities on all paths completes the tree. To find the conditional probability of major given poor, we need from the seventh path on the tree. Also, can be obtained by adding the probabilities for all paths including a poor rating. . Thus

**Exercise 3**

- Two observations:
- shooting in the air is the same as missing,
- if
*A*and*B*are both alive when*C*shoots, it is better for*C*to kill*B*than*A*.

- If
*A*shoots in the air

- If
*A*shoots at*B*

- If
*A*shoots at*C*

*A*to shoot in the air. And respectively, ; ; ; .

**Exercise 4**

**a.**- We must take into account not only the number of deductions, but also the frequencies. We could simply use the definition of the mean. Add up 0 201 times, 1 287 times, and so on, then divide by the total frequency. Or we could regard the data as grouped. In either case

By calculator,

*s*=1.834. Because there are 1533 observations, it would make virtually no difference whether we divided by*n*or*n*-1. In fact, calculations show , calculated by dividing by*n*.Recall that about 68 of the data should fall within one standard deviation of the mean. The range is 0.643 to 4.311. It includes the values 1, 2, 3, and 4 and contains 1134 observations. This is 1134/1533=0.740 or 74 of the data. The Empirical Rule works surprisingly well, despite the skewness of the data.

**b.**- There is 1533 forms, of which 1332 claim at least 1 deduction.

We note that 287+364+332=983 forms claim at least 1 and at most 3 deductions, so

Alternatively, there are 1332 forms with at least one deduction, of which 983 claim at most three deductions. Therefore,

**Exercise 5**

**a.**- Because the firm holds 30
of the market, we assume that the probability of winning any one bid is 0.30. The company can win
bids. To have
*Y*come out equal to 3, the company must win all the bids.

Similarly,

The event*Y*=1 can occur in three different ways, corresponding to the choice of which of the three bids wins.

Similarity,

We assumed that the probability of winning was the 0.3 on all three bids and that whether or not one bid won didn't change the probability that any other bid would win. If there are no systematic difference among the bids, we can't think of any reason why the probability would change. Barring collusion, there's no obvious reason for dependence, either. The assumptions seem at least fairly reasonable.

**b.**-

and

**c.**- The expected value shouldn't change. On average, the company
should still win 30
of all its bids, and 30
of 3 is 0.9, as we found
in the the previous exercise. What should change is the variance. The
statement of the exercise indicates that there's higher probability that the
company will either win all its bids or lose all its bids, that is, that
*Y*will come out either 3 or 0. Therefore, there must be a smaller probability that Y will come out 1 or 2. If there is higher probability on the more extreme values, the variance must be larger.

**Exercise 6**

**a.**- The value of
*Y*_{1}depends on the number of Reds on four consecutive rolls. This number has a binomial distribution. The probability distribution of*Y*_{1}is shown below:*Y*_{1}0 16 32 48 64 1/16 4/16 6/16 4/16 1/16 **b.**- We lose only if we get five Reds in a row, otherwise we win 1 dollar. The probability distribution of
*Y*_{2}is shown below:*Y*_{2}1 33 1/32 31/32 **c.**- First criterion: Expected Value. Both have expectation of 32, so same on this criterion.
Second criterion: Median. Median of

*Y*_{1}is 32 but median of*Y*_{2}is 33, so*Y*_{2}is better.