Practice Problems on Combinatorics, Probability and Statistics

Problem 1: Forming a committee from a group of women and men

I. From a group of 5 women and 7 men, how many different committees consisting of 2 women and 3 men can be formed? What if 2 of the men are feuding and refuse to serve on the committee together? 

2. A dance class consists of 22 students, of which 10 are women and 12 are men. If 5 men and 5 women are to be chosen and then paired off, how many results am possible? 

3. Consider 3 urns: • Um A contains 2 white balls and 4 red balls. • Um B contains 8 white balls and 4 red balls. • Um C contains 1 white ball and 3 red balls. 

What is the probability that the ball chosen from um A was white given that exactly 2 balls were selected? 

4. Medical case histories indicate that different illnesses may produce identical symptoms. Suppose a particular set of symptoms, which we will denote as event H, occurs only when any one of the three illnesses- a; B; or c-occurs. (For sake of simplicity, we will assume that illnesses A, a, and c are mutually exclusive.) Studies show these probabilities of getting the three illnesses: 

P(A) = 0.01 P(B) = 0.005 P(c) = ff02 

The probabilities of developing the symptoms H, given a specific illness, are 

P(HIA) = 0.90 P(81B) = 0.95 P(81c) = 0.75 

Assuming that an ill person shows the symptoms H, what is the probability that the person has illness a? 

5. A salesman has scheduled two appointments to sell encyclopaedias. His first appointment will lead to a sale with probability .3, and his second will lead independently to a sale with probability .6. Any sale made is equally likely to be either for the deluxe model, which costs $1000, or the standard model, which costs $500. Determine the probability mass function of X, the total dollar value of all sales. 

6. A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective. Find the expected number of defective items in the sample. 

7. When coin 1 is flipped, it lands on heads with probability .4; when coin 2 is flipped, it lands on heads with probability .7. One of these coins is randomly chosen and flipped 10 times. 

(a) What is the probability that the coin lands on heads on exactly 7 of the 10 flips? (b) Given that the first three of these ten flips lands heads, what is the conditional probability that exactly 7 of the 10 flips land on heads? 

8. People enter a gambling casino at a rate of 1 every 2 minutes. 

(a) What is the probability that no one enters between 12:00 and 12:05? (b) What is the probability that at least 4 people enter the casino during that time? 

9. Bratwurst lengths in Bayern have a mean of 22.4cm, with a standard deviation of 2.4cm. If the Bratwurst lengths in Bayern are normally distributed, what is the approximate probability that more than 5 out of 250 have a length of more than 28cm?