For two statistically independent events A, B such that P(B) = 2P(A) and
P(A∪ B) = 5/8 , find the probability of P(A) .
The Police have found the blood of the jewel thief near the hotel safe. 10% of
all women belong to the blood group and 2% of all men. 30% of the hotel staff
are women. Assuming that this was an inside job, what is the probability that
the thief was a woman?
Ten balls are tossed into four boxes so that each ball is equally likely to fall into
any box. What is the probability function for the number of balls in the last
box?
Assume that the distribution of the heights of adult males is approximately
normal with a mean of 69 ins and a standard deviation of 3 ins. What is the
conditional probability that an individual will be taller than 72 ins if it is known
that he is taller than 70 ins.?
Suppose two random variables X and Y have the joint probability density
function f (x, y) = x + y with 0 ≤ x ≤1, 0 ≤ y ≤1 . What is the conditional
mean E(Y | X =1)?
Following previous question, find the conditional variance var (Y | X =1).
A man runs twice a week. The average duration of one of his outings is 40
minutes with a standard deviation of 5 minutes. His average running time per
week is 1 hour 20 minutes with a standard deviation of 4 minutes. Given that
he ran for only 15 minutes on Monday, what is the expected duration of his
Friday outing?
Imagine that a random sample of size 160 is drawn from a normal population
where the standard deviation is 25 in order to test the null hypothesis that the
mean is 104 against the alternative that it is 100. Calculate the probability of a
Type II error in a one-tailed test with a 5% significance level.
Following question (12), what sample size is required for the probability of the
Type II error to be 0.05?
On a windless day, the coordinates of a parachutist's landing point, measured
relative to two axes passing at right angles through the target centre, are
distributed as independent normal variates with a mean of zero and a standard
deviation of 9 metres. The target area is surrounded by a circular pebble pit of
radius 15 metres. Calculate the probability that the parachutist will land on the
pebbles.
Following question (14), what is the least amount of rope required to enclose
an area into which he can be 90% confident of falling?
The levels of personal bank accounts are distributed normally with a mean of
$690 and standard deviation of $90. What is the probability that an individual
will have more than $780 if it is known that he has more than $720?
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