First, make sure you can do all of the problems on the first exam, as well as the review problems for the first exam.

**Probability / Counting / Binomial Distribution**

- Two dice are rolled. Find the probability that:
- both numbers are 1.
- The first die rolls a 3 and the second rolls a 2
- There is one 3 and one 2 (in any order)
- The numbers on the two dice add to 7
- Both numbers are even
- One number is even and one is odd.

- Calculate by hand, showing all your work:
- 6!
- C(10, 4) (“10 choose 4”)
- P(10, 4)

- An urn contains 3 red marbles, 4 blue marbles and 2 green marbles. Two marbles are randomly picked without replacement. Find the probability that both marbles are green.
- A medical condition affects 5% of the population. A medical test for this condition has been developed. In a random sample of 1000 people, we find the following results:

Condition Present Condition Absent Total Test + 49 95 144 Test – 1 855 856 Total 50 950 1000 - Find the probability of a person testing positive, given that they have the condition.
- Find the probability of a person testing positive, given that they do not have the condition.
- Find the probability of a person having the condition, given that they test positive.
- Find the probability of a person having the condition, given that they test negative.

- For the following probability distribution, sketch a graph of the distribution and calculate its expected value and standard deviation:

x 0 1 2 3 P(x) .3 .4 .15 .15 - A fair coin is tossed 4 times. Sketch a graph of the binomial distribution for this scenario.
- A coin is biased so that the probability of getting heads is 25%. The coin is flipped 10 times. Find the probability of getting more than 1 heads.

Also do the following:

Chapter 5 Review (page 234-235) #12, 16

Chapter 6 Review (page 282) #6, 7, 9

**Normal Distribution / Sampling Distributions**

- Let z have the standard normal distribution. For the following probabilities, draw the normal curve, shade the appropriate region and determine the value (between 0 and 1):
- P(z < 0.5)
- P(-1.25 < z < 1.5)
- P(z > 2)

- Let x be a normally distributed random variable with a mean of 50 and standard deviation of 5. For each of the following, draw the normal curve, shade the appropriate region, and find the probabilities:
- P(40 < x < 50)
- P(40 < x < 60)
- P(45 < x < 55)
- P(45 < < 55), if a random sample of size n = 25 is taken.

- The lifetime of a certain kind of TV has a normal distribution with a mean of 120 months and a standard deviation of 5 months.
- What is the probability that a randomly selected TV of this type would have a lifetime of between 100 and 120 months?
- The manufacturer of this TV wishes to offer a full guarantee on all TVs which last fewer than a certain number of months. If they wish to refund only 5% of TVs, for how many months should they guarantee their TVs?
- For a random sample of 4 TVs, what is the probability that the average lifetime of TVs in that sample is between 100 and 120 months?

Also do the following:

Chapter 7 Review (page 356) #11, 16, 18, 20, 21