Practice Questions: Independent events and Bayes Theorem
1.
If events A and B are such that they are independent and P( A)  0.3 , P ( B )  0.5 , find
P ( A  B ) and P ( A  B ) . Are events A and B mutually exclusive?
2
The probability that a bus from A to B will leave on time is 0.7 and the probability that
it will leave on time and also arrive on time is 0.56. What is the probability that if such
a bus leaves on time it will also arrive on time?
3. Rachel and Anna play each other at badminton. Each game results in either a win for Rachel
or a win for Anna. The probability of Rachel winning the first game is 0.6. If Rachel wins a
particular game, the probability of her winning the next game is 0.7, but if she loses, the
probability of her winning the next game is 0.4.
(i)
Find the conditional probability that Rachel wins the first game, given that she
loses the second
[5]
Find the probability that Rachel wins 2 games and loses 1 game out of the first
three games they play.
[4]
(ii)
4.
The people living in 3 houses are classified as children (C), parents (P) or grandparents
(G). The numbers living in each house are shown in the table below.
House number 1
House number 2
House number 3
4C, 1P, 2G
(i)
(ii)
(iii)
2C, 2P, 3G
1C, 1G
All the people in all 3 houses meet for a party. One person at the party is chosen
at random. Calculate the probability of choosing a grandparent.
Ans: 3/8
A house is chosen at random. Then a person in that house is chosen at random.
Using a tree diagram, or otherwise, calculate the probability that the person
chosen is a grandparent.
Ans: 17/42
Given that the person chosen by the method in part (ii) is a grandparent, calculate
the probability that there is also a parent living in the house.
Ans: 10/17
5. A standard die is rolled twice ( remember that rolling a standard die is equivalent to
selecting an integer from
{ 1, 2, 3, 4, 5, 6 } uniformly at random.)
Let A be the event that the first roll is 1, 2, 3 or 4.
Let B be the event that the first roll is even.
Let C be the event that the sum of the two rolls is 11 or 12.
(i)
Write down Pr(A), Pr(B), and Pr(C).
(ii)
Write down Pr(A intersect B), Pr(A intersect C), and Pr(B intersect C).
(iii)
Write down Pr(A U C) and Pr(B U C).
(iv)
Write down whether A and B are independent events.
(v)
Write down whether A and C are independent events.
(vi)
Write down whether B and C are independent events.
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