PROBABILITY AND STATISTICS, A16, TEST 1

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1 PROBABILITY AND STATISTICS, A16, TEST 1 Name: Student number (1) (1.5 marks) i) Let A and B be mutually exclusive events with p(a) = 0.7 and p(b) = 0.2. Determine p(a B ) and also p(a B). ii) Let C and D be independent events with p(c) = 0.7 and p(d) = 0.8. Determine p(c D).

2 2 PROBABILITY AND STATISTICS, A16, TEST 1 (2) (1.5 marks) Diseases I and II are prevalent among people in a certain population. It is known that 10% of the population will contract disease I sometime in their lifetime, 15% will contract disease II eventually, and 3% will contract both diseases. Determine the probability that a randomly chosen person from this population will contract both diseases, given that he or she has contracted at least one disease.

3 PROBABILITY AND STATISTICS, A16, TEST 1 3 (3) (1.5 marks) A manufacturer of consumer electronics product expects 2% of units to fail during the warranty period. A random sample of 600 units is tracked for warranty performance. Let X be the random variable which counts the number of products which fail during the warranty period in the sample. i) What is the expected value of X and what is the standard deviation? ii) Estimate the probability that precisely 12 units will fail?

4 4 PROBABILITY AND STATISTICS, A16, TEST 1 (4) (2 marks) An insuarance company offers its policy holders a number of different premium payment options. For a randomly selected policyholder, let X be the number of months between successive payments. The cumulative distribution function of X is as follows 0 x < x < 3 F (x) = x < x < x i) Compute p(2 X 6) and p(x 4). ii Compute the mean and the standard deviation of X.

5 PROBABILITY AND STATISTICS, A16, TEST 1 5 (5) (2 marks) An automobile insuarance company divides cutomers into three categories, good risks, medium risks, and poor risks. Assume that 70% of the customers are good risks, 20% are medium risks, and 10% are poor risks. Assume that during the course of a year, a good risk customer has probability of filing an accident claim, a medium risk customer has probability 0.01, and a poor risk customer has probability A customer is chosen at random. i) What is the probability that the customer is a good risk and has not filed a claim? ii) Given that the customer has filed a claim, what is the probability that the customer is a good risk?

6 6 PROBABILITY AND STATISTICS, A16, TEST 1 (6) (1.5 marks) A batch contains 420 bacteria 224 of which are not capable of cellular replication. If you take a random sample of 32 bacteria from the batch: i) What is the expected value and the standard deviation of the number of bacteria not capable of replication in the sample? ii) Estimate the probability that precisely 16 of the bacteria in the sample are not capable of replication.

7 PROBABILITY AND STATISTICS, A16, TEST 1 7 (7) (2 marks) Every day, a weather forecaster predicts whether or not it will rain. For 80% of rainy days, she correctly predicts that it will rain. For 90% of non-rainy days, she correctly predicts it will not rain. Suppose that 15% of days are rainy and 85% are non-rainy. i) If this forecaster predicts tomorrow will be a rainy day, what is the probability that it will actually rain? ii) If this forecaster predicts tomorrow will be a non-rainy day, what is the probability that it will actually rain? iii) What is the probability that the forecaster will be correct seven days in a row?

8 8 PROBABILITY AND STATISTICS, A16, TEST 1 (8) (2 marks) Three people toss a fair coin and the odd one pays for coffee. If the coins turn up the same, they are tossed again. i) What is the expected value and the standard deviation of the number tosses needed to resolve who pays for coffee? ii) Find the probability that fewer than 4 tosses are needed.

9 PROBABILITY AND STATISTICS, A16, TEST 1 9 (9) (1.5 marks) A state has a 5/35 lottery game. The ticket costs 1$. Matching all 5 numbers (jackpot) pays 200,000 dollars, matching 4 of the 5 numbers pays 250$ and mathching 3 of the 5 numbers pays 2$. The rest of the tickets are non-winning. What is the expected value of this game and what is the standard deviation?

10 10 PROBABILITY AND STATISTICS, A16, TEST 1 Relevant Formulas f(x) = n C x p x (1 p) n x, E(X) = np, V ar(x) = np(1 p). f(x) = x 1 C r 1 p r (1 p) x r, E(X) = r r(1 p), V ar(x) =. p p 2 f(x) = k C x N k C n x NC n, E(X) = np, V ar(x) = np(1 p) N n N 1, p = k N. f(x) = e λ λ x, E(X) = λ, V ar(x) = λ. x!

MATH1215: Mathematical Thinking Sec. 08 Spring Worksheet 9: Solution. x P(x)

MATH1215: Mathematical Thinking Sec. 08 Spring Worksheet 9: Solution. x P(x) N. Name: MATH: Mathematical Thinking Sec. 08 Spring 0 Worksheet 9: Solution Problem Compute the expected value of this probability distribution: x 3 8 0 3 P(x) 0. 0.0 0.3 0. Clearly, a value is missing