Mixed Probability Questions
1. Two events A and B are such that: P A = 158 , P B  13 , P A B  15
Calculate the probabilities that:
a) both events occur
b) only one of the two events occurs
c) neither event occurs
2. Exactly 60% of a school class are boys. 90% of the boys and 75% of the girls each buy one
raffle ticket, the rest buying none. Calculate the probability that the wining ticket will be bought
by a boy.
3. Doctors believe that a virus infects three people in every thousand in the population. A test has
been devised which is not perfect, but gives a positive result for 95% of those who have the
virus. It also gives a positive result for 2% of those who do not have the virus. Suppose that
someone selected at random from the population takes the test and it gives a positive result.
Calculate the probability that the person really has the virus.
4. Events C and D are such that
4
1
P(C ) = , P(C  D) =
and P(C D) = 5
14
7
3
Find : a) P(C  D) b) P(D) and c) P(D C)
5. Two cards are drawn at random from a pack of 7 cards numbered 1,2,3,4,5,6,7. Find the
probability that:
a)
the product of the numbers is even
b) at least one prime number is included
c)
exactly one prime number is included
d) the product of the numbers is even and exactly one of the numbers is prime
e)
HINTS
1.
2.
3.
4.
5.
the product of the numbers is even given that exactly one of the numbers is prime
Use multiplication formula to calculate (a), then draw a Venn diagram
Draw a tree diagram, with first girls/boys then buy/not buy on the branches
Draw a tree diagram with virus/not virus then tests positive/tests negative on the branches
Draw a Venn diagram to work out (a), and then use the multiplication formula to get (b). Then use your Venn diagram to get (c)
Consider all the possible ways the outcome can be achieved in each case; for example for (a) we can have first number odd/second
number even or first number even/second number anything. For (e) use multiplication rule
Solutions to Mixed Probability Questions
1
11
1
1. (a)
(b)
(c)
15
15
5
5
5.
(a)
(b)
7
2. 0.643
3. 0.125
6
7
(c)
4. (a)
4
7
(d)
5
21
(b)
3
7
(e)
2
3
(c)1
3
4