. a) We select two balls at random without replacement from a basket that
contains four white and eight black balls.
i) Compute the probability that both balls are white (2 marks)
ii) Compute the probability that the second ball is white
(3 marks)
b) Prove that by Chebyshev’s inequality, if x is any random variable with the
mean E(x)=µ, a variance V (x)=s 2 and ? be a small positive deviation
form the mean, then ( ) ( )
?2
p x-µ =? =V x . (10 marks)
c) If ?=5 and k =5 in (b) above, what is the probability of a deviation from
the mean by more than k standard deviations (5 standard deviations)?
(4 marks)

2. Prove that;
a) If E1 and E2 are subsets of S, then;
( ) ( ) ( ) ( ) 1 2 1 2 1 2 P E ?E =P E + P E -P E nE (12 marks)
b) if events 1 2 E and E are subsets of S such that 1 E is a subset of 2 E then
( ) ( ) 1 2 p E < p E (10 marks)
3. a) A manufacturer of metal pistons finds that on the average, 12% of his
pistons are rejected because they are either oversize or undersize. What is
the probability that a bath of 10 pistons will contain;
i) No more than 2 rejects? (5 marks)
ii) at least 2 rejects? (4 marks)
b) A company pays its employees an average wage of Kshs 3.25 an hour with
a standard deviation of 60 cents. If the wages are approximately normally
distributed, determine;
i) The proportion of the workers getting wages between Kshs 2.75
and Kshs 3.69 an hour (6 marks)
ii) The minimum wage of the highest 5% (2 marks)

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