Download Mathematics Department, NUI Galway

Mathematics Department, NUI Galway
MA235: Probability
Problems Sheet 2 : Probability Rules
1. State the three axioms which must be satisfied by any probability function.
If A and B are any two events defined on a sample space, use the above axioms to prove
that
P (A ∩ B) ≥ P (A) + P (B) − 1
Hence, if P (A) = 0.5 and P (B) = 0.9 determine the range of possible values of P (A ∩ B).
2. (a) Prove in general for any three events that
P (A1 ∪ A2 ∪ A3 ) = P (A1 ) + P (A2 ) + P (A3 ) − P (A1 ∩ A2 )
− P (A1 ∩ A3 ) − P (A2 ∩ A3 ) + P (A1 ∩ A2 ∩ A3 )
(b) Consider an experiment in which a fair six-sided die is rolled three times. Let A1 = {1
or 2 on the first roll}, A2 = {3 or 4 on the second roll}, and A3 = {5 or 6 on the
third roll}. It is given that P (Ai ) = 1/3, i = 1, 2, 3; P (Ai ∩ Aj ) = (1/3)2 , i = j; and
P (A1 ∩ A2 ∩ A3 ) = (1/3)3 .
Use part (a) to show that P (A1 ∪ A2 ∪ A3 ) = 1 − (1 − 1/3)3 .
3. If sixteen married couples were involved in an accident in which only six people survived,
calculate the probability that there is at least one married couple amongst the survivors.
4. Each of n sticks is broken into one long and one short part. The 2n parts are randomly
regrouped into n pairs, from each of which a new stick is formed. Find the probability
that:
(a) all the parts are joined as they were originally;
(b) all the long parts are joined with all the short parts.
When cells are exposed to harmful radiation, some chromosones break and behave like
these sticks. The ‘long’ side of the chromosone is the part that contains the centromere.
If any two ‘long’ or any two ‘short’ sides unite, the cell dies. What is the probability that
a cell containing ten chromosones dies when exposed to radiation?
5. (Banach’s matchbox problem) The Polish mathematician Banach kept two match boxes,
one in each pocket. Each box initially contains n matches. Whenever he wanted a match
he reached out at random into one of his pockets. When he found that the box he picked
was empty what was the probability that there were k, (0 ≤ k ≤ n), matches left in the
other box? [Hint: divide the problem into two cases according as the left or right box is
empty.] Hence, or otherwise, show that
n
2n − k
1
=1
n
22n−k
k=0
6. A fair coin is tossed until the first time that the same result appears on two consecutive
tosses. Show that the probability of doing this in exactly n tosses is 1/2n−1 , n ≥ 2. Hence
find the probability that:
(a) the experiment ends before the fifth toss;
(b) an even number of tosses is required.
7. Bus tickets in a certain city contain four numbers U, V, W and X. Each of these numbers
is equally likely to be any of the ten digits 0, 1, . . . , 9, and the four numbers are chosen
independently. A bus rider is said to be lucky if U + V = W + X. What proportion of
riders are lucky?
8. Bean seeds from supplier A have an 85% germination rate and those from supplier B have
a 75% germination rate. A seed packaging company purchases 40% of their bean seeds
from supplier A and 60% from supplier B and mixes these seeds together.
(a) Find the probability that a seed selected at random from the mixed seeds will germinate.
(b) Given that a seed germinates, find the probability that the seed was purchased from
supplier A.
9. Suppose that
(a) an aircraft engine will fail, when in flight, with probability 1 − p;
(b) engines fail independently;
(c) an aircraft will make a successful flight if at least 50 percent of its engines remain
operative.
For what values of p is a two-engine plane preferable to a four-engine plane?
[p < 2/3]
On a particular route two-engine planes are used twice as often as four-engine planes. If
p = 0.9 and Aunt Agatha arrived safely on this route, what is the probability that she
came on a four-engine plane?
[0.3347]
10. In his garden last year, Mr Riley planted 51 onion sets of which he knew that 50 were
from accredited stock, and 1 was a small old onion thrown in by his mischievous wife. He
had no knowledge of which was which, and it turned out that one onion failed to grow.
Assuming that the probability of accredited stock growing is 0.9, compared with 0.4 for
small old onions thrown in by mischievous wives, find the probability that the failed onion
was, in fact, not from accredited stock.
11. Consider the following version of the game of craps: The player rolls two dice. If the
sum on the first roll is 7 or 11, the player wins the game immediately. If the sum on the
first roll is 2, 3 or 12 the player loses the game immediately. However, if the sum on the
first roll is 4, 5, 6, 8, 9 or 10, then the two dice are rolled again and again until the sum
is either 7 or the original value. If the original value is obtained a second time before 7
obtained, then the player wins. If 7 is obtained before the original value is obtained a
second time, then the player loses. Determine the probability that the player will win
this game.
[0.493]
12. The probability that any child in a certain family will have blue eyes is 1/4, and this
feature is inherited independently by different children in the family.
(a) If there are five children in the family and it is known that at least one of these
children has blue eyes, what is the probability that at least three of the children
have blue eyes?
(b) If it is known that the youngest child in the family has blue eyes, what is the
probability that at least three of the children have blue eyes?
(c) Explain why the answers in parts (a) and (b) are different.