Problem Set 3 (Probability)

1) Roll two dice (for the list of outcomes, see my lecture notes)

Let A be the event "The sum is 7

Let Be be the event "The second number is odd"

a) Calculate P(A) and P(B)

b) Calculate P(A and B)

c) Calcuate P(A or B) directly

d) Calculate P( A or B) using the Addition Rule

e) Calculate the conditional probabilities P(A|B) and P(B|A)

f) Are A, B mutually exclusive: (Explain)

g) Are A, B independent?

2) Out of 100 females, 60 responded that they were vegetarians. Out of 150 males, 20 responded that they were vegetarians

Randomly choose one person out of the above population

Let M stand for males, F for females, V for vegetarians, notV for not vegetarians

a) Construct a two - way table (contingency table)

b) Calculate P(M), P(F), P(V) and P(notV)

c) Calcualte P (M and V),

d) Calculate P(M or V)

e) Calculate P(M|V), P(V|M) and P(notV|F)

f) Are the events M and notV mutually exclusive?

g) Are the above two events independent?

h) Which two of the 4 events are mutually exclusive?

3) Choose randomly one student out of the frequency table:

GradeFrequency

A3

B5

C7

D6

F4

a) Probability that a selected student got B

b) Probability that the student passed (D or better)

c) Probability that that the student got lower than B, given that the student got higher than D?

4) Suppose that you know:

P(A) = .2

P(B) = .3

P(A or B) = .4

a) Calculate P(notA)

b) P(A and B)

c) P(A | B)

5) A family with two children have the following sample space:

S = {GG, GB, BG, BB} ....the first is the older. We assume that the probability of boy is 60%

Calculate the probability that a family with two children has:

a) Two girls

b) Two girls given that they have at least one girl

6) A doctor, on the average, has 70% success rate in making a correct diagnosis. Three people will visit her tomorrow to get their diagnosis. Calculate the probability that:

a) All patients will get the correct diagnosis

b) Nobody will be diagnosed correctly.

c)At least one patient will get correct diagnosis

d) At most two patients will get correct diagnosis.