MATH1715 Tutee Revision Questions

Question: 26/12/2010
The probability that a certain type of inoculation takes effect is 0.995. Using a Poisson
approximation or otherwise, compute the probability that at most 2 out of 400 people given
the inoculation will find that it has not taken effect.
Response: Let X be the number for whom the inoculation does not take effect. For n = 400
people, X ∼ Bin(n = 400, p = 0.005). What is the approximate distribution satisfied by
X? The question requires P{X ≤ 2}.
Question: 26/12/2010
Three factories manufacture 20%, 30% and 50% of computer chips a company sells. If
the fractions of defective chips are 0.4%, 0.3% and 0.2%, respectively, what fraction of the
defective chips come from the third factory?
Response: Let D denote defective, and A, B, C the source factory. Thus we are told, for
example, P{A} = 0.20 and P{D|A} = 0.004.
We require P{C|D}. Thus use Bayes’s theorem.
P{C|D} =
P{D|C} P{C}
P{D|A} P{A} + · · · + P{D|C}P{C} .
Question: 26/12/2010
A box contains 3 blue balls and 5 red balls. Ann and Bob take turns (with Ann going first)
to draw a ball from the box, without replacement, until a blue ball is drawn. What is the
probability that Ann will draw the blue ball first?
Response: Draw a tree diagram is one way. On the first stage Ann either draws blue (with
probability 3/8) or red (with probability 5/8). If Ann draws red on stage one, on the second
stage Bob either draws blue (with probability 3/7) or red (with probability 4/7). If Bob
draws red on stage two, on the third stage Ann either draws blue or red, and so on. You
require all routes which end with Ann drawing blue.
The question can be answered in other ways. For example, let Ar denote Ann’s draw on
stage r, r = 1, 3, 5, 7, and Bs denote Bob’s draw on stage s, s = 2, 4, 6, 8.
P{Ann wins} = P{A1 = blue} + P{(A1 = red) ∩ (B2 = red) ∩ (A3 = blue)} + · · ·
where P{A1 = blue} = 3,
and 8
and so on.
P{(A1 = red) ∩ (B2 = red) ∩ (A3 = blue)} = 5 4 3
× ×
8 7 6 ,
Question: 26/12/2010
In the Euro millions lottery, a player must choose 5 correct digits from {1, 2, . . ., 50} on the
main board and two “lucky stars” from the nine digits 1-9. The lottery draw yields a “5+2”
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