Page 78 - Probability and Statistical Inference

1. Notions of Probability 55

1.4.16 Consider the digits 0, 1, ..., 9. Use these digits at random to form a
four (five) digit number. Then, find the probability of forming
(i) a four digit random number, not starting with a zero, which
would be an even number while each digit appears exactly once;
(ii) a four digit random number which would be an even number
where the digits can be repeated, and starting with a zero is
allowed;
(iii) a five digit random number, not starting with a zero, which would
be divisible by the number five while each digit appears only
once.
1.4.17 In a twin engine plane, we are told that the two engines (#1, #2)
function independently. We are also told that the plane flies just fine when at
least one of the two engines are working. During a flying mission, individually
the engine #1 and #2 respectively may fail with probability .001 and .01.
Then, during a flying mission, what is the probability that the plane would
crash? The plane would complete its mission?
1.4.18 Suppose that A , ..., Ak are disjoint events. Let B be another event.
1
Then, show that

1.4.19 Suppose that A , A are events. Then, show that
1 2

1.4.20 Suppose that A , A are disjoint events. Then, show that
1 2
(i) P(A | )= P(A )/{1 – P(A )} if P(A ) ≠ 1;
1 1 2 2
(ii) P(A | A ∪ A ) = P(A )/{P(A ) + P(A )}.
1 1 2 1 1 2
1.5.1 Suppose that a random variable X has the following pmf:
X values: –2 0 1 3 8
Probabilities: .2 p .1 2p .4

where p ∈ (0, .1].
(i) Is it possible to determine p uniquely?
(ii) Find P{|X – .5| > 2} and P{|X – .5| = 2.5}.

1.5.2 Suppose that a random variable X has the following pmf:
X values: –2 0 1 3 8
Probabilities: .2 p .1 .3 – p .4