1. A card is drawn from a standard pack of 52. The following events are defined.
R: the card is red.
H: the card is a heart
K: the card is a King
B: the card is blacka. Name two complementary events.
b. Name two mutually exclusive events.
c. Name two events which are not independent.
d. Show that K and H are independent.
|
2. Assume the events S = (it is raining in Sydney) and W = (it is raining in Wellington) are independent, and that P(S) = 0.2 and P(W) = 0.3. If the Prime Minister of New Zealand rings Sydney to check the weather in Australia, use Venn diagrams to calculate the probability that:
|
 |
3. Assume the events S = (it is raining in Sydney) and W = (it is raining in Wellington) are independent, and that P(S) = 0.2 and P(W) = 0.3. If the Prime Minister rings Auckland to check the weather, use the probability rules and formulae to calculate the probability that:
a. It is raining in at least one of the two cities.
b. It is raining in Sydney but not in Wellington.
4. (i) Give the meanings of the terms:
a. Independent
b. Disjoint for events A and B.
(ii) Can two events be both independent and disjoint? (give reasons)
5. A school contains 365 students. Assuming that all days are equally likely to be chosen and ignoring leap years, what is the probability that a student chosen at random has a birthday on Christmas day or New Year's day?
6. Suppose that A and B are events, with P(A) = 0.18, P(B) = 0.4 and P(A 
B) = 0.06.
(i) Are A and B independent? Give a reason for your answer.
(ii) Are A and B mutually exclusive? Give a reason for your answer.
(iii) Find P(A 
B)
7. A school heating plant has a main system and a back up system. Both are subject to random breakdowns; the main system operates for only 80% of the time, and at least one of the two systems is operational for 95% of the time; both systems are operational for 45% of the time.
Let A be the event "main system is operational" and B be the event "back-up system is operational".
(i) Complete the statement P(A B) = P(A) +______.
Hence find the proportion of the time that the back-upsystem is operational.
(ii) Show that the two systems are NOT statistically independent in their operation.
8. In a secondary school, all Year 12 and Year 13 students studying physics were asked about their use of calculators in physics. The following table summarises the information that was obtained.
|
Takes physics
|
Owns a calculator
|
Regularly uses a calculator
|
|
|
Year 12
|
158
|
120
|
84
|
|
Year 13
|
164
|
134
|
128
|
(i) If a Year 12 student who takes physics is randomly selected, what is the probability that he or she regularly uses a calculator.
(ii) If a Year 13 student who takes physics is randomly selected, find the probability that he or she owns a calculator but does not use it regularly.
(iii) If a person from either year level who takes physics is chosen randomly, find the probability that he or she does not own a calculator.
