1 Let X be the number of heads in three tosses of a fair coi

1) Let X be the number of heads in three tosses of a fair coin. What is the probability function of X ?

2) 5 patients enter a medical clinic which has 9 different doctors. If each patient is equally likely to see any of the doctors, what is the probability that no two patients see the same doctor?

3) 36 identical candies are distributed randomly to 4 children. What is the probability that each child receives at least 5 candies?

4) A certain math class consists of 24 males and 18 females. 8 students are chosen from this class at random. What is the probability that precisely 4 of those chosen will be females?

5) Let Y be a random variable with values 1,2,3.

If P (Y = 1) = .3, P (Y = 2) = .1, what is P (Y = 3)? What is P (Y 2)?

6) For the random variable Y of the last problem, what is E(Y ) and V (Y )?

7) Let P (A) = .4, P (B) = .6 and P (A B) = .8. Find P (A B). Are A

and B independent? Explain.

8) 80% of our students are right handed while the remainder are left handed.

30% of right right handers wear glasses, whereas 60% of left handers wear glasses. What is the probability that a randomly chosen student wears glasses?

9) 80% of our students are right handed while the remainder are left handed.

30% of right right handers wear glasses, whereas 60% of left handers wear glasses. If a student wears glasses, what is the probability that he is left handed?

10) If X is a random variable with E(X ) = 42 and V (X ) = 6, use Tchebysh- eff ’s theorem to find a lower bound on the probability that 36 < X < 48.

1) Let X be the number of heads in three tosses of a fair coin. What is the probability function of X ?

2) 5 patients enter a medical clinic which has 9 different doctors. If each patient is equally likely to see any of the doctors, what is the probability that no two patients see the same doctor?

3) 36 identical candies are distributed randomly to 4 children. What is the probability that each child receives at least 5 candies?

4) A certain math class consists of 24 males and 18 females. 8 students are chosen from this class at random. What is the probability that precisely 4 of those chosen will be females?

5) Let Y be a random variable with values 1,2,3.

If P (Y = 1) = .3, P (Y = 2) = .1, what is P (Y = 3)? What is P (Y 2)?

6) For the random variable Y of the last problem, what is E(Y ) and V (Y )?

7) Let P (A) = .4, P (B) = .6 and P (A B) = .8. Find P (A B). Are A

and B independent? Explain.

8) 80% of our students are right handed while the remainder are left handed.

30% of right right handers wear glasses, whereas 60% of left handers wear glasses. What is the probability that a randomly chosen student wears glasses?

9) 80% of our students are right handed while the remainder are left handed.

30% of right right handers wear glasses, whereas 60% of left handers wear glasses. If a student wears glasses, what is the probability that he is left handed?

10) If X is a random variable with E(X ) = 42 and V (X ) = 6, use Tchebysh- eff ’s theorem to find a lower bound on the probability that 36 < X < 48.

Solution

1.

This is a binomial distribution with n = 3, p = 0.5. Thus,

P(x) = 3Cx (0.5)^x (0.5)^(3-x)

P(x) = 3Cx (0.5)^3

P(x) = 0.125 (3Cx) or 0.125 (3!/[x!(x-3)!]) [ANSWER]

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2.

There are 5^9 ways for the patients to choose doctors.

There are 9P5 = 15120 ways to choose 5 different doctors for 5 patients.

Thus,

P(all different) 15120/5^9 = 0.00774144 [answer]

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