F71SM STATISTICAL METHODS
Tutorial on Section 2 PROBABILITY
1. (a) BywritingA∪B∪CasA∪DwhereD=B∪C,verifythat
P(A∪B∪C) = P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C)
(b) BywritingA∩B∩C asD∩C ,whereD=A∩B,verifythat P(A ∩ B ∩ C) = P(A)P(B|A)P(C|A ∩ B)
2. Consider three events A, B, C which are such that A and B are mutually exclusive, A and C are independent, B and C are independent, P(A) = 0.3, P(B) = 0.6, and P(C) = 0.5.
(a) Find the probabilities of the events (i) A ∪ (B ∩ C), and (ii) (A ∪ B) ∩ C. [i) 0.6, ii) 0.45]
(b) Are events A, B, C independent?
3. In a group of 150 adult males, 100 play the National Lottery, 30 bet on horses, 20 do both, and 40 do neither. Suppose a person is chosen at random from the group. Let A = selected person bets on horses and B = selected person plays the National Lottery.
Are events A and B independent? Are they mutually exclusive?
4. Five students are selected at random, with replacement, from a group of 20 students of
whom 12 are male and 8 are female.
What is the probability that the first student selected is male? [0.6] What is the probability that the third student selected is male? [0.6] What is the probability that the fifth student selected is male? [0.6]
5. In a small-scale lottery, a player selects 4 different numbers from the list 1,2,3,4,5,6,7,8,9. The promoter independently also selects 4 different numbers and a further different ‘bonus’ number from the same list. The player wins the jackpot prize if his 4 num- bers match the 4 main numbers selected by the promoter, and he wins a runner-up prize if he can match 3 of the main numbers and the ‘bonus’ number.
Calculate the probabilities that a player wins (i) the jackpot prize, and (ii) a runner-up prize. [i) 0.0079, ii) 0.0317]
6. A group of four life policies is selected at random from a collection of ten policies on male lives and five policies on female lives. Calculate the probability that the group selected contains at least two policies on male lives, given that it contains at least one policy on a life of each sex. [0.9130]
7. In a certain constituency, 30% of voters are ‘blue collar’ workers, of whom 46% voted Conservative at the last election. Of the remaining voters, 36% voted Conservative. A voter is selected at random from the population of voters and it turns out this person voted Conservative. Calculate the probability that the selected voter is a ‘blue collar’ worker. [0.3538]
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8. A motor insurer classifies accidents leading to claims into three categories. Category 1, 2, and 3 accidents make up 30%, 60%, and 10% of the total respectively. The probability that a category 1 accident results in loss of life is 0.01 and that it results in non-fatal serious injury is 0.1. The corresponding probabilities for a category 2 accident are 0.1 and 0.3, and for a category 3 accident are 0.4 and 0.6.
Draw a tree diagram and find:
(a) the probability that a claim does not relate to an accident involving loss of life or serious injury; [0.627]
(b) the probability that a claim which involved loss of life arose out of a category 3 accident. [0.3883]
9. A and B play a game in which A starts. A throws two fair six-sided dice and wins the game if the total score is exactly 5. If A does not win, play passes to B, who throws three fair six-sided dice and wins if the total score is exactly 7, otherwise A throws again and so on. The game continues until either A throws 5 and wins or B throws 7 and wins. Show that the probability for A to win the game is 9/14.
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