Islamic University of Gaza
Department of Math.
Second semester 13/5/2009
‫االسم‬
Q1
Q2
‫الرقم‬
7
6
Probability Theory
First midterm exam
Time: one hour
Q3 Q4 Q5 Q6 Total
6
6
9
6
40
‫المدرس‬
Solve the following questions:
Q1) A box contains 5 red balls and 3 white balls. A sample of 4 balls is drawn at
random and without replacement from the box.
i) Determine the probability mass function of the number of red balls in the
sample.
ii) What is the probability that there are at least two red balls in the sample.
iii) Determine the mean and variance of the number of red balls in the sample.
1
Q2)(a) If P (A | B )  0.7, P (A )  0.5 and P (B )  0.2.
i) Are A and B independent events? Explain.
ii) Determine P (B | A ) .
iii) Determine P (A  B ) .
Q3) Suppose that the probability mass function of a random variable X is
x
f (x )
0
0.125
a
f (a )
2
0.375
(i)If E (X )  1.5 , find a and f (a ) .
(ii) Determine the cumulative distribution function of X .
2
3
0.125
Q4) An assembly consists of two parts. Suppose the probability that the first and
the second part is defective equals 0.05 and 0.10 respectively. Assume that the
parts are independent. Determine the probability mass function of the number
of defective parts in the assembly.
Q5) The probability that a sample of water is contaminated equals 0.1. If the
samples are independent.
i) If 20 samples of water are checked, what is the probability that more than one
sample is contaminated?
3
ii) What is the probability that 10 samples must be checked to find the first
contaminated sample?
iii) What is the mean number of the samples that must be checked to obtain
three contaminated samples?
Q6) A company buys semiconductors from two suppliers. 30% of the
semiconductors are from supplier I and the remaining are from supplier II. If
85% of the semiconductors from supplier I and 95% of the semiconductors from
supplier II meet the specifications.
i) If a semiconductor is selected at random, what is the probability that it meets
the specifications?
ii) If a semiconductor meets the specifications, what is the probability that it is
from supplier I?
4