Problem 1. Suppose a deck of 52 cards is shuffled and the top two cards are dealt.

 

a)      How many ordered pairs of cards could possibly result as outcomes?

 

Assuming each of these pairs has the same chance, please calculate:

b)      The chance that the first card is an ace;

c)      The chance that the second card is an ace;

d)      The chance that both cards are aces;

e)      The chance of at least one among the two cards are aces.

 

Problem 2.

Suppose there are two electrical components. The chance that the first component fails is 10%. If the first component fails, the chance that the second component fails is 20%. But if the first component works, the chance that the second component fails is 5%. Please calculate the probabilities of the following events:

a)      At least one of the components works;

b)      Exactly one of the components works;

c)      The second component works.

 

Problem 3.

Assume that boys and girls are born with equal frequencies, not quite true, but the difference doesn't matter for this problem. Consider families of exactly two children. Given that a family has a boy, what is the probability that both children are boys?

 

Problem 4. Drawing tickets with and without replacement

With replacement

A box contains tickets marked 1, 2, ¡­, 10. A ticket is drawn at random from the box. Then this ticket is replaced in the box and a second ticket is drawn at random. Find the probabilities of the following events:

a)      The first ticket drawn is number 1 and the second ticket is number 2.

b)      The numbers on the two tickets are consecutive integers (the first number drawn is one less than the second number drawn).

c)      The second number drawn is bigger than the first number drawn.

 

Without replacement

d)      Repeat a) through c) above assuming instead that the first ticket drawn is not replaced, so the second ticket drawn must be different from the first.

 

Problem 5.

Suppose that 1% of the population has a particular disease. There is a test on a blood sample which yields one of two results, positive or negative. It is found that 95% of the people having the disease produce a positive result. But 2% of the people without the disease will also produce a positive result. What is the probability that a person randomly chosen from the population will have the disease, given that the person's test was positive?

 

Problem 6. The odds against an event occurring are 10 to 1. What is the chance of the event? What if the odds were 5 to 1 against?

 

Problem 7. Counting TAT in random sequences.

We randomly extracted a 20k non-coding sequence from E. Coli. Genome and found the distribution of nucleotides A, C, G, T in that region:

Pr(A)=0.2, Pr(C)=0.3, Pr(G)=0.3, Pr(T)=0.2.

 

a)    Please simulate 500 i.i.d. DNA sequences of length 20k each using the above distribution and count the number of occurrences of TAT in each simulated sequence. Plot the Histogram of the counts you have obtained, and compare the Histogram to the Normal distribution with same mean and standard deviation. (Note: i.i.d. stands for identically and independently distributed)

b)    If we have observed 250 occurrence of TAT in a DNA sequence of length 20K, is TAT significantly over-representative in that sequence (under the hypothesis that the DNA sequence is i.i.d. and following the distribution described above)?

 

Problem 8.

To pass a test you have to perform successfully two consecutive tasks, one easy and one hard. The easy task you think you can perform with probability z, and the hard task you think you can perform with probability h<z. You are allowed three attempts, either in the order (easy, hard, easy) or in the order (hard, easy, hard). Whichever order, you must be successful on two out of the three attempts in a row to pass. Assuming that your attempts are independent, in what order should you choose to take the tasks in order to maximize your probability of passing the test?

 

Problem 9. True/False. Explain your answer in details.

a). If EX=10, then E(3-2X)=17.

b). If Var(X)=2, then Var(4X+3)=11.

c). A six-sided die is rolled once. Let the random variable X denote the number on the upturned face. Define A={X<6}, and B={X is odd}. If the die is fair, P(A|B)=1.

d). The events A and B in the previous problem are independent.

e). A fair coin is tossed until we observe heads exactly 50% of the time, then we stop. Let X be the total number of heads we observe before we stop. X is a Binomial random variable.

 

Problem 10.

Two fair dice are rolled independently. Let X be the maximum of the two rolls, and Y the minimum.

a)    What is P(X = x) for x = 1,..,6?

b)    What is P(Y = y| X = 3) for y = 1,¡­,6?

c)     What is the joint distribution of X and Y?

d)    What is E(X+Y)?

 

Problem 11.

A teaching assistant gives a quiz to his section. There are 10 questions on the quiz and no part credit is given. After grading the papers, the TA writes down for each student the number of questions the student got right and the number wrong. The average number of right answers is 6.4 with an SD of 2.0; the average number of wrong answers is 3.6 with the same SD of 2.0. Then what is the correlation coefficient of the two numbers?

 

Problem 12.

An investigator collected data on height and weight of college students; results can be summarized as follows.

                                               Average                 SD

--------------------------------------------------------------

                  Men¡¯s height                 70 inches               3 inches

                  Men¡¯s weight                144 pounds      21 pounds

                  Women¡¯s height            64 inches               3 inches

                  Women¡¯s weight           120 pounds      21 pounds

 

The correlation coefficient between height and weight for the men was about 0.60; for the women, it was about the same. If you take the men and the women together, the correlation between the height and weight would be

      a) just about 0.60         b) somewhat lower       c) somewhat higher      d) can not tell

Choose one option, and explain.

 

Problem 13.

Pearson and Lee obtained the following results in a study of about 1,000 families:

      Average height of husband is about 68 inches, SD is about 2.7 inches

      Average height of wife is about 63 inches, SD is about 2.5 inches

      The correlation r is about 0.25

 

Predict the height of a wife when the height of her husband is

      (a) 72 inches          (b) 64 inches    (c) 68 inches          (d) unkown

Predict the height of a husband when the height of his wife is

      (a) 58 inches          (b) 68 inches    (c) 63 inches          (d) unkown

 

Problem 14.

One kind of plant has only blue flowers and white flowers. According to Mendel's genetic model, the offsprings of a certain cross have a 75% chance to be blue-flowering (blue is a dominant trait), and a 25% chance to be white-flowering, independently of each other. Two hundred seeds of such a cross are raised, and 142 turn out to be blue-flowering. Are the data consistent with Mendel's law? Answer yes or no, and explain why.

 

Problem 15.

One large course has 900 students, broken down into section meetings with 30 students each. The section meetings are led by teaching assistants. On the final, the class average is 63, and the SD is 20. However, in one section the average is only 55. The TA argues this way:

If you took 30 students at random from the class, there is a pretty good chance they would average below 55 on the final. That is what happened to me ¨C chance variation.

Is this a good defense? Answer yes or no, and use hypothesis testing to explain your answer. (You can assume the final scores follow a normal curve)