HW 1.8 More Probability
1. A card is drawn at random from a standard deck.
a) Find the probability that the card is an ace.
b) Find the probability that the card is black.
c) Find the probability that the card is an ace and black.
d) Find the probability that the card is an ace or black.
2. The events A and B are such that P(A) =0.5, P(B) = 0.7, and P(A∩B) = 0.2. Find
a) P(A  B)
b) P(B’)
c) P(A’∩B)
3. The events A and B are such that P(A) = 0.35, P(B) = 0.5 and P(A∩B) = 0.15. Making use of a Venn
diagram (where appropriate) find:
a) P(A’)
b) P(A  B)
c) P(A  B’)
4. A coin is tossed three times.
a) Draw a tree diagram and from it write down the sample space.
Use the results of part (a) to find the probability of obtaining
b) only one tail
c) 2 tails in succession
5. In a class of 25 students it is found that 6 of the students play both tennis and chess, 10 play tennis only,
and 3 do not participate in any activities at all. A student is selected at random from this group. Using a
Venn diagram, find the probability that the student:
a) plays both tennis and chess
b) plays chess only
c) does not play chess
6. A blue and a red die are rolled together (both numbered one to six).
a) Draw a lattice diagram that best represents this experiment.
b) Find the probability of observing an even number with the red die.
c) Find the probability of observing a sum of 7 or an odd number on the red die.
7. A card is drawn at random from a standard deck of 52 playing cards. Find the probability that the card
drawn is
a) a diamond
b) a club or spade
c) a black card or a picture card
d) red or a queen
8. A and B are two events such that P(A) = p, P(B) = 2p and P(A∩B) = p2 .
a) Given that P(A  B) = 0.4, find p.
Hence, use a Venn Diagram to find the following:
b) P(A’  B)
c) P(A’∩B’)
9. In a group of 30 students 20 hold an Australian passport, 10 hold a Malaysian passport and 8 hold both
passports. The other students hold only one passport (that is neither Australian nor Malaysian). A
student is selected at random.
a) Draw a Venn diagram which describes this situation.
b) Find the probability that this student has both passports.
c) Find the probability that the student holds neither passport.
d) Find the probability that the student holds only one passport.
HW 1.9 Conditional Probability
1. Two events A and B are such that P(A) = 0.6, P(B) = 0.4, and P(A∩B) = 0.3, find the probability of the
following events:
a) A  B
b) A│B
c) B│A
d) A│B’
2. A and B are two events such that P(A) = 0.3, P(B) = 0.5 and P(A  B) = 0.55. Find the probability of
the following events:
a) A│B
b) B│A
c) A│B’
d) A’│B’
3. A box contains 5 red, 3 black, and 2 white cubes. A cube is randomly drawn and has its colour noted.
The cube is then replace, together with 2 more of the same colour. A second cube is then drawn.
a) Find the probability that the first cube selected is red.
b) Find the probability that the second cube selected is black.
c) Given that the first cube selected was red, what is the probability that the second cube selected is
black?
4. Two unbiased coins are tossed together. Find the probability that they both display heads given that at
least one is showing a head.
5. A money box contains 10 discs, 5 of which are yellow, 3 of which are black and 2 green. Two discs are
selected in succession, the first disc not being replaced before the second is selected.
a) Draw a tree diagram representing this process.
b) Hence, find the probability that the discs will be of a different color.
c) Given that the second disc was black, what is the probability that both were black?
6. Two dice are rolled. Find the probability that the faces are different given that the dice show a sum
of 10.
7. Given that P(A) = 0.6 and P(B) = 0.7, and that A and B are independent events. Find the probability of
the event
a) A  B
b) A∩B
c) A│B'
d) A'∩B
8. The probability that an animal will still be alive in 12 years is 0.55 and the probability that its mate will
still be alive in 12 years is 0.60. Find the probability that
a) both will still be alive in 12 years
b) only the mate will still be alive in 12 years.
c) at least 1 will still be alive in 12 years.
d) the mate is still alive in 12 years given that only one is still alive in 12 years.
9. Tony has a 90% chance of passing his math test while Tanya has an 85% chance of passing the same
test. If they both sit for the test, find the probability that
a) only one of them passes
b) at least one passes the test
c) Tanya passed given that at least one passed.
10. The probability that Rory finishes a race is 0.55 and the probability that Millicent finishes the same
race is 0.6. Because of team spirit, there is an 80% chance that Millicent will finish the race if Rory
finishes the race. Find the probability that
a) both will finish the race
b) Rory finishes the race given that Millicent finishes.