How to Pick Binomial Trials

This post provides additional practice for the ideas discussed in this blog post Picking Two Types of Binomial Trials.

Problem 1
Suppose there are two basketball players, each makes 50% of her free throws. In one game, player A attempted 10 free throws and player B attempted 15 free throws. Assume that the free throws of each player are independent of each other. Suppose you are told that in this game, 8 of their free throws were hits. Given this information:

1. What is the probability that player A made 4 of the hits?
2. What is the mean number of hits made by player A?
3. What is the variance of the number of hits made by player A?
4. What is the probability that player B made 5 of the hits?
5. What is the mean number of hits made by player B?
6. What is the variance of the number of hits made by player B?

Problem 2
A student took two multiple choice statistics quizzes that were independent of each other, i.e., results of one quiz did not affect the results on the other. One quiz had 8 questions and the other quiz had 10 questions. Each question had 5 choices and only one of the choices was correct. The student did not study. So she answered each question by random guessing. If the student was told that she had 5 correct answers in the two quizzes:

1. What is the probability that the student answered 3 or more questions correctly in the first quiz?
2. What is the mean number of correct answers in the first quiz?
3. What is the variance of the number of correct answers in the first quiz?
4. What is the probability that the student answered at most 3 questions correctly in the second quiz?
5. What is the mean number of correct answers in the second quiz?
6. What is the variance of the number of correct answers in the second quiz?

Refer to this post to find the background information.

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Problem 1

$\displaystyle (1) \ \ \ \ \frac{286650}{1081575}=0.265$

$\displaystyle (2) \ \ \ \ \frac{3461040}{1081575}=3.2$

$(3) \ \ \ \ 1.36$

$\displaystyle (4) \ \ \ \ \frac{360360}{1081575}=0.33318$

$(5) \ \ \ \ 4.8$

$(6) \ \ \ \ 1.36$

Problem 2

$\displaystyle (1) \ \ \ \ \frac{3276}{8568}=0.3824$

$\displaystyle (2) \ \ \ \ \frac{19040}{8568}=2.22$

$\displaystyle (3) \ \ \ \ \frac{1300}{1377}=0.944$

$\displaystyle (4) \ \ \ \ \frac{6636}{8568}=0.7745$

$\displaystyle (5) \ \ \ \ \frac{23800}{8568}=2.78$

$\displaystyle (6) \ \ \ \ \frac{1300}{1377}=0.944$