Statistical Methods Assignment No. 1

1. Sanjit, Prashant and Bharath have probabilities 0.8, 0.7 and 0.6 to solve a given

problem. If the problem is solved what is the probability that (i) only Sanjit could solve it, (ii) only Prashant could solve it, (iii) only Bharath could solve it ?

2. In a modeling agency 2/3 of the models are under 22 years of age. Also 3/5 of the

models are male and 5/8 of the models are female or 22 years of age or older. What is the probability that a model selected at random from this agency is a female and under 22 years of age?

3. The figures within the boxes in the diagram of an electronics system show the

probabilities of the corresponding system components to function. What is the probability that the entire system functions?

4. A machine to detect improper welds in a fabricating shop detects 80% of all improper

welds, but it also incorrectly indicates an improper weld on 5%of all satisfactory welds. Past experience indicates that 10% of all welds are improper. What is the probability that a weld which the machine indicates to be defective is, in fact satisfactory.

5. On a flight from Boston to Calcutta my luggage did not arrive with me. It was

transferred thrice on account of change in flight and the probabilities that the transfer was not done in time were estimated to be 4/10, 2/10 and 1/10 respectively in order of transfer. What is the probability that the first airline goofed?

6. Suppose that in a certain casino there are three types of slot machines in equal

numbers with pay-off frequencies 1/3, 2/3, 2/3 respectively. One of these machines paid off twice in four cranks. What is the probability of a pay-off on the next crank?

7. Show that n

1i

n

1iii .)1n()P(A)AP(

= =

8. In answering a question on a multiple choice test a student either knows the answer

or she guesses. Let p be the probability that she knows the answer and 1 - p the probability that she guesses. Assume that a student who guesses at the answer will be

.8

.9

.9

.8

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correct with probability 1/m, where m is the number of multiple - choice alternatives. What is the conditional probability that a student knew the answer to a question given that she answered it correctly?

9. A blood test is 95% effective in detecting a certain disease, when it is, in fact present.

The test also yields a false positive result for 1% of the healthy persons tested. If 0.5% of the population actually has the disease, what is the probability that a person has a disease given that his test result is positive?

10. If P(Ac) = 0.3, P(B) = 0.4 and P(ABc) = 0.5, find P(B / ABc). 11. Four computer firms A, B, C, D are bidding for a certain contract. A survey of past

bidding success of these firms on similar contracts shows the following probabilities of winning : P(A) = 0.35, P(B) = 0.15, P(C) = 0.3, P(D) = 0.2. Before the decision is made to award the contract, firm B withdraws its bid. Find the new probabilities of winning the bid for A, C and D.

12. A teaset has four cups and saucers with two cups and saucers in each of two different

colours. If the cups are placed at random on the saucers, what is the probability that no cup is on a saucer of the same colour ?

13. Ram rolls two fare dice. If the sum of the numbers shown is 7 or 11 he wins, if it is

2,3 or 12 he loses. If it is any other number j, he continues to roll two dice, until the sum is j or 7, whichever is sooner. If it it 7 he loses, if it is j he wins. What is the probability that Ram wins.

14. An electric circuit looks as in the figure below, where the numbers indicate the

probabilities of failure for the various links, which are all independent. What is the probability that the circuit is closed ?

1/5 1/5

A 1/3 B 1/4 1/4 15. The probability that a family has n children is pn, for n = 1,2,.. A child can be a

boy or a girl with equal probability. Find the probability that a family has at least a boy. Given that a family has at least one boy, what is the conditional probability that it has two or more boys?

16. A speaks truth 3 times out of 4 and B 7 times out of 10. They both assert that a white

ball is drawn from a bag containing 6 balls of different colours ( one is white). Find the probability of truth of assertion.