Probability and Statistics
Name__________________________

Spring 1998 Flex-Mode and Flex-Time 45-733

Practice Midterm

Keith Poole

(10 Points)

1. You roll one dice three times. If it is known that a 1 appeared at least once in the three rolls, what is the probability that a 1 appeared only once?Clearly

because you have to take into account the three possibilities for the "1" to appear. Hence

Probability and Statistics
Name__________________________

Spring 1998 Flex-Mode and Flex-Time 45-733

Practice Midterm

Keith Poole

(10 Points)

2. We have 3 urns. One contains 5 red balls, 3 white, and 2 blue. A second urn contains 4 red, 4 white, and 2 blue. And the third urn contains 1 red, 7 white, and 2 blue. What is the probability that the three balls are the same color.

(5/10)(4/10)(1/10) + (3/10)(4/10)(7/10) + (2/10)(2/10)(2/10) =

(20 + 84 + 8)/1000 = .112

Probability and Statistics
Name__________________________

Spring 1998 Flex-Mode and Flex-Time 45-733

Practice Midterm

Keith Poole

(10 Points)

3. Suppose we have the distribution function:

**F(x)** =

Find the corresponding discrete probability distribution, **f(x)**.

To

In any event, note that:

Now, since

{ 2/52 x = 1 { { 6/52 x = 2 { { 8/52 x = 3 { f(x) = { 10/52 x = 4 { { 12/52 x = 5 { { 14/52 x = 6 { { 0 otherwise

Spring 1998 Flex-Mode and Flex-Time 45-733

Practice Midterm

Keith Poole

(10 Points)

4. Suppose we have a continuous probability distribution**f(x)** =

a. Find c.

òb. Find **E(X)**.

c. Find **VAR(X)**.

Hence,

Spring 1998 Flex-Mode and Flex-Time 45-733

Practice Midterm

Keith Poole

(10 Points)

5. Suppose we have a discrete bivariate probability distribution

**f(x,y)** =

Find **VAR(X)** and **VAR(Y)**

y 1 2 3 ------------ 1 | 3 2 1 | 6 | | x 2 | 7 6 5 |18 | | 3 |11 10 9 |30 | | --------------- 21 18 15 |54

Spring 1998 Flex-Mode and Flex-Time 45-733

Practice Midterm

Keith Poole

(10 Points)

6. Suppose we have two groups of people. In the first group there are 9 women and 10 men and in the second group there are 13 women and 5 men. Suppose a person is randomly selected from each group. Let **Y** = 1 if the two persons are the same sex, and let **Y** = 0 if the two persons are not the same sex. What is the probability distribution of **Y**?

Hence

{ .5117 y = 0 { f(y) = { .4883 y = 1 { { 0 otherwise

Spring 1998 Flex-Mode and Flex-Time 45-733

Practice Midterm

Keith Poole

(10 Points)

7. A woman fires 10 shots at a target. The probability is .9 that she will hit the target on any given shot. The shots are independent of one another. What is the probability that she has hit the target at least twice if it is known that she has hit the target at least once.

Note that this is a binomial distribution with parameters[1 - .1

Spring 1998 Flex-Mode and Flex-Time 45-733

Practice Midterm

Keith Poole

(10 Points)

8. Suppose we have the continuous function

**f(x)** =

Does a c exist such that this is a probability distribution.

òNow, since

Spring 1998 Flex-Mode and Flex-Time 45-733

Practice Midterm

Keith Poole

(10 Points)

9. Suppose we have 100 executives and we have to form 4 committees of 37, 23, 15, and 10 people, respectively. How many ways can the committees be selected.This is the Multinominal Coefficient:

(100 choose 37 23 15 10 15) = 100!/37!23!15!10!15!

Spring 1998 Flex-Mode and Flex-Time 45-733

Practice Midterm

Keith Poole

(10 Points)

10. Suppose we have the bivariate continuous probability distribution

**f(x,y)** =

Find **P(X > 1/2 Ç
Y < 1/2)**

(2/3)ò

(2/3)[(x