Question

1. The probability that a student has a Visa card (event V) is 0.30. The probability that a student has a MasterCard (event M) is 0.40. The probability that a student has both cards is 0.12. (1) Find the probability that a student has either a Visa card or a MasterCard. (2) In this problem, are V and M independent? Why?

2.

This is a contingency table describes 100 business students.

	Gender
Major	Female(F)	Male(M)
Accounting (A)	22	28
Economics(E)	16	14
Statistics(S)	12	8

Find each probability.

(1) P(A)                

(2) P(A?M)      

(3) P(A|F)           

(4) P(F|S)           

(5) P(E?F)

3.

Suppose that 12% of the women who purchase over-the-counter pregnancy testing kits are actually pregnant. For a particular brand of kit, if a woman is pregnant, the test will yield a positive result 96% of the time and a negative result 4% of the time. If she is not pregnant, the test will yield a positive result 5% of the time and a negative result 95% of the time.
Suppose the test comes up positive.

Let event A be Positive test.

Let event B be Pregnant.

(1) P(A|B) =?

(2) P(A|B')=?

(3) What is the probability that she is really pregnant?
(i.e. What is P(B|A) (the probability that she is really pregnant under the condition that the test comes up positive)?)

4.

Consider a random experiment “tossing un-fair coin two times” where the probability of head for the unfair coin is 0.4. Let random variable X be the number of heads. That is, X follows the binomial distribution with parameters n = 2, ?=0.4.

(1) Calculate P(X?1).

(2) Calculate P(X>1).

(3) Find the expected value of X.

(4) Find the standard deviation of X.

Answer 1

Ans:

1)

Given that

P(V)=0.3

P(M)=0.4

P(V and M)=0.12

a)P(V or M)=P(V)+P(M)-P(V and M)

=0.3+0.4-0.12

=0.58

b)P(V and M)=0.12

P(V)*P(M)=0.3*0.4=0.12

As,P(V and M)=P(V)*P(M),V and M are independent.

2)

i)P(A)=(22+28)/100=0.5

ii)P(A and M)=28/100=0.28

iii)P(A/F)=P(A and F)/P(F)=22/(22+16+12)=0.44

IV)P(F/S)=P(F and S)/P(S)=12/(12+8)=0.6

V)P(E or F)=P(E)+P(F)-P(E and F)

P(E)=(16+14)/100=0.3

P(F)=(22+16+12)/100=0.5

P(E and F)=16/100=0.16

So,

P(E or F)=0.3+0.5-0.16=0.64