Probability and Random Variables Rapid Review for AP Statistics

Review the following concepts if necessary:

• Probability for AP Statistics
• Random Variables for AP Statistics
• Normal Probabilities for AP Statistics
• Simulation and Random Number Generation for AP Statistics
• Combining Random Variables for AP Statistics

Rapid Review

1. A bag has eight green marbles and 12 red marbles. If you draw one marble from the bag, what is P(draw a green marble)?

Answer: Let s = number of ways to draw a green marble.

Let f = number of ways to draw a red marble.



2. A married couple has three children. At least one of their children is a boy. What is the probability that the couple has exactly two boys?

Answer: The sample space for having three children is {BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG}. Of these, there are seven outcomes that have at least one boy. Of these, three have two boys and one girl. Thus, P(the couple has exactly two boys they have at least one boy) = 3/7.

3. Does the following table represent the probability distribution for a discrete random variable?



Answer: No, because

4. In a standard normal distribution, what is P(z > 0.5)?

Answer: From the table, we see that P(z < 0.5) = 0.6915. Hence, P(z > 0.5) = 1 – 0.6915 = 0.3085. By calculator, normalcdf (0.5,100) = 0.3085375322.

5. A random variable X has N(13,0.45). Describe the distribution of 2 – 4X (that is, each datapoint in the distribution is multiplied by 4, and that value is subtracted from 2).

Answer: We are given that the distribution of X is normal with μX = 13 and σX = 0.45. Because μ_{a ± bX} = a ± bμ_X, μ_{2 – 4X} = 2 – 4μ_X = 2 – 4(13) = –50. Also, because σ_{a ± bX} = bσ_X, σ_{2 – 4X} = 4σ_X = 4(0.45) = 1.8.