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# Probability and Random Variable Free Response Practice Problems for AP Statistics

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By — McGraw-Hill Professional
Updated on Feb 3, 2011

Review the following concepts if necessary:

### Problems

1. Find μX and σX for the following discrete probability distribution:
2. Given that P(A) = 0.6, P(B) = 0.3, and P(B|A) = 0.5.
1. P(A and B) = ?
2. P(A or B) = ?
3. Are events A and B independent?
3. Consider a set of 9000 scores on a national test that is known to be approximately normally distributed with a mean of 500 and a standard deviation of 90.
1. What is the probability that a randomly selected student has a score greater than 600?
2. How many scores are there between 450 and 600?
3. Rachel needs to be in the top 1% of the scores on this test to qualify for a scholarship. What is the minimum score Rachel needs?
4. Consider a random variable X with μX = 3, σ2X = 0.25. Find
1. μ3 + 6X
2. σ3 + 6X
5. Harvey, Laura, and Gina take turns throwing spit-wads at a target. Harvey hits the target 1/2 the time, Laura hits it 1/3 of the time, and Gina hits the target 1/4 of the time. Given that somebody hit the target, what is the probability that it was Laura?
6. Consider two discrete, independent, random variables X and Y with μX = 3, σ2X = 1, μY = 5, and σ2Y = 1.3. Find μXY and σXY.
7. Which of the following statements is (are) true of a normal distribution?
1. Exactly 95% of the data are within two standard deviations of the mean.
2. The mean = the median = the mode.
3. The area under the normal curve between z = 1 and z = 2 is greater than the area between z = 2 and z = 3.
8. Consider the experiment of drawing two cards from a standard deck of 52 cards. Let event A = "draw a face card on the first draw," B = "draw a face card on the second draw," and C = "the first card drawn is a diamond."
1. Are the events A and B independent?
2. Are the events A and C independent?
9. A normal distribution has mean 700 and standard deviation 50. The probability is 0.6 that a randomly selected term from this distribution is above x. What is x ?
10. Suppose 80% of the homes in Lakeville have a desktop computer and 30% have both a desktop computer and a laptop computer. What is the probability that a randomly selected home will have a laptop computer given that it has a desktop computer?
11. Consider a probability density curve defined by the line y = 2x on the interval [0,1] (the area under y = 2x on [0,1] is 1). Find P(0.2 ≤ X ≤ 0.7).
12. Half Moon Bay, California, has an annual pumpkin festival at Halloween. A prime attraction to this festival is a "largest pumpkin" contest. Suppose that the weights of these giant pumpkins are approximately normally distributed with a mean of 125 pounds and a standard deviation of 18 pounds. Farmer Harv brings a pumpkin that is at the 90% percentile of all the pumpkins in the contest. What is the approximate weight of Harv's pumpkin?
13. Consider the following two probability distributions for independent discrete random variable X and Y:
14. If P(X = 4 and Y = 3) = 0.03, what is P(Y = 5)?

15. A contest is held to give away a free pizza. Contestants pick an integer at random from the integers 1 through 100. If the number chosen is divisible by 24 or by 36, the contestant wins the pizza. What is the probability that a contestant wins a pizza?
16. Use the following excerpt from a random number table for questions 15 and 16:

17. Men and women are about equally likely to earn degrees at City U. However, there is some question whether or not women have equal access to the prestigious School of Law. This year, only 4 of the 12 new students are female. Describe and conduct five trials of a simulation to help determine if this is evidence that women are under represented in the School of Law.
18. Suppose that, on a planet far away, the probability of a girl being born is 0.6, and it is socially advantageous to have three girls. How many children would a couple have to have, on average, until they have three girls? Describe and conduct five trials of a simulation to help answer this question.
19. Consider a random variable X with the following probability distribution:
1. Find P(X ≤ 22).
2. Find P(X > 21).
3. Find P(21 ≤ X < 24).
4. Find P(X ≤ 21 or X > 23).
20. In the casino game of roulette, a ball is rolled around the rim of a circular bowl while a wheel containing 38 slots into which the ball can drop is spun in the opposite direction from the rolling ball; 18 of the slots are red, 18 are black, and 2 are green. A player bets a set amount, say \$1, and wins \$1 (and keeps her \$1 bet) if the ball falls into the color slot the player has wagered on. Assume a player decides to bet that the ball will fall into one of the red slots.
1. What is the probability that the player will win?
2. What is the expected return on a single bet of \$1 on red?
21. A random variable X is normally distributed with mean μ, and standard deviation σ (that is, X has N(μ,σ)). What is the probability that a term selected at random from this population will be more than 2.5 standard deviations from the mean?
22. The normal random variable X has a standard deviation of 12. We also know that P(x > 50) = 0.90. Find the mean μ of the distribution.

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