PROBABILITY
30.23 A pointPis chosen at random on the circlex^2 +y^2 = 1. The random variable
Xdenotes the distance ofPfrom (1,0). Find the mean and variance ofXand
the probability thatXis greater than its mean.
30.24 As assistant to a celebrated and imperious newspaper proprietor, you are given
the job of running a lottery, in which each of his five million readers will have an
equal independent chance,p, of winning a million pounds; you have the job of
choosingp. However, if nobody wins it will be bad for publicity, whilst if more
than two readers do so, the prize cost will more than offset the profit from extra
circulation – in either case you will be sacked! Show that, however you choose
p, there is more than a 40% chance you will soon be clearing your desk.
30.25 The number of errors needing correction on each page of a set of proofs follows
a Poisson distribution of meanμ. The cost of the first correction on any page is
αand that of each subsequent correction on the same page isβ. Prove that the
average cost of correcting a page is
α+β(μ−1)−(α−β)e−μ.30.26 In the game of Blackball, at each turn Muggins draws a ball at random from a
bag containing five white balls, three red balls and two black balls; after being
recorded, the ball is replaced in the bag. A white ball earns him $1, whilst a
red ball gets him $2; in either case, he alsohas the option of leaving with his
current winnings or of taking a further turn on the same basis. If he draws a
black ball the game ends and he loses all he may have gained previously. Find
an expression for Muggins’ expected return if he adopts the strategy of drawing
up tonballs, provided he has not been eliminated by then.
Show that, as the entry fee to play is $3, Muggins should be dissuaded from
playing Blackball, but, if that cannot be done, what value ofnwould you advise
him to adopt?
30.27 Show that, for larger, the value at the maximum of the PDF for the gamma
distribution of orderrwith parameterλis approximatelyλ/
√
2 π(r−1).
30.28 A husband and wife decide that their family will be complete when it includes
two boys and two girls – but that this would then be enough! The probability
that a new baby will be a girl isp. Ignoring the possibility of identical twins,
show that the expected size of their family is
2(
1
pq− 1 −pq)
,
whereq=1−p.
30.29 The probability distribution for the number of eggs in a clutch is Po(λ), and the
probability that each egg will hatch isp(independently of the size of the clutch).
Show by direct calculation that the probability distribution for the number of
chicks that hatch is Po(λp) and so justify the assumptions made in the worked
example at the end of subsection 30.7.1.
30.30 A shopper buys 36 items at random in a supermarket, where, because of the
sales tax imposed, the final digit (the number of pence) in the price is uniformly
and randomly distributed from 0 to 9. Instead of adding up the bill exactly,
she rounds each item to the nearest 10 pence, rounding up or down with equal
probability if the price ends in a ‘5’. Should she suspect a mistake if the cashier
asks her for 23 pence more than she estimated?
30.31 Under EU legislation on harmonisation, all kippers are to weigh 0.2000 kg, and
vendors who sell underweight kippers must be fined by their government. The
weight of a kipper is normally distributed, with a mean of 0.2000 kg and a
standard deviation of 0.0100 kg. They are packed in cartons of 100 and large
quantities of them are sold.
Every day, a carton is to be selected at random from each vendor and tested