# Data and Probability

1. How many variations in first-, second-, and third-place finishes are possible in a 100-yd dash with 6 runners?

2. In a Chinese restaurant, the menu lists 8 items in Column A and 6 items in column B. To order a dinner, the diner is told to select 3 items from column A and 2 from column B. How many dinners are possible?

3. Suppose a family plans 6 children, and the probability that a particular child is a girl is 1/2. Find the probability that the 6 child family has exactly 2 girls.

4. Suppose 2 cards are drawn without replacement from an ordinary deck of 52. Find the probability that both cards are aces.

5. You pay $6 to play a game where you will roll a die, with a payoff as follows: $8 for a 6, $7 for a 5, and $4 for any other result. What are your expected winnings? Is the game fair?

6. Find the mean for: 105, 108, 110, 115, 106, 110, 104, 113, 117.

7. Find the median and mode (or modes) for: 32, 35, 36, 44, 46, 46, 59.

8. Find the range and standard deviation for: 14, 17, 18, 19, 32.

9. Find the following area under the standard normal curve: between z=-1.88 and z=2.10.

10. Find a z-score such that 8% of the area under the curve is to the right of z.

EC. A machine that fills quart orange juice bottles is set to fill them with 32.1 oz. If the actual contents of the cartons vary normally, with a standard deviation of .1 oz, what percent of the cartons contain less than a quart (32 oz).

#### Solution Preview

1. How many variations in first-, second-, and third-place finishes are possible in a 100-yd dash with 6 runners?

The first place can be filled in 6 ways.

The second place can then be filled in 6 -1 = 5 ways.

The third place can then be filled in 5 – 1 = 4 ways.

Thus, the total number of variations is 6 x 5 x 4 = 120 ways.

2. In a Chinese restaurant, the menu lists 8 items in Column A and 6 items in column B. To order a dinner, the diner is told to select 3 items from column A and 2 from column B. How many dinners are possible?

The 3 items from column A can be picked in 8C3 ways (i.e. 8 choose 3). Note that order is not important in this case.

This value is = 8!/5!3! = 56 ways where ! means factorial.

The 2 items from column B can be picked in 6C2 ways (i.e. 6 choose 2). Note that order is not important in this case.

This value is 6!/4!2! = 15 ways.

Thus, we can pick 3 items from column A and 2 items in column B in 56 x 15 = 840 ways.

3. Suppose a family plans 6 children, and the probability that a particular child is a girl is 1/2. Find the probability that the 6 child family has exactly 2 girls.

The number of girls follows a binomial distribution with probability p = 0.5 and sample size n = 6.

The probability formula for exactly k girls is given by:nCk pk(1 – p)n-k

In this case k = 2. Thus, probability for 2 girls is 6C20.520.56-2

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#### Solution Summary

This provides examples of a variety of topics in data and probability, including counting, expected value, measures of central tendency, and z-score.