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MATH 240 HOMEWORK I
Due date: 5th week class time for each section
1. Suppose traffic engineers have coordinated the timing of two traffic lights to encourage a run of green lights. In particular, the timing was designed so that with probability 0.8 a driver will find the second light to have the same color as the first. Assuming the first light is equally likely to be red or green,
(a) what is the probability P(G2) that the second light is green?
(b) what is P(W), the probability that you wait for at least one light?
(c) what is P(G1 | R2), the conditional probability of a green first light given a red second light?
2. A manufacturing plant makes radios that each contain an integrated circuit (IC) supplied by three sources A, B, and C. The probability that the IC in a radio came from one of the sources is 1/3, the same for all sources. ICs are known to be defective with probabilities 0.001, 0.003, and 0.002 for sources A, B, and C, respectively.
(a) What is the probability that any given radio will contain a defective IC?
(b) If a radio contains a defective IC, find the probability it came from source A.
3. You have two biased coins. Coin A comes up heads with probabaility ¼. Coin B comes up heads with probability ¾. However, you are not sure which is which so you choose a coin randomly and you flip it. If the flip is heads, you guess that the flipped coin is B; othervise you guess that the flipped coin is A. Let events A and B designate which coin was picked. What is the probability P(C) that your guess is correct?
4. a) A pair of fair dice is rolled ten times. Find the probability that ‘eight’ will show at least once.
b) A pair of fair dice are rolled. What is probability that the second die lands on a higher value than does the first?
5. (a) Assuming drivers are independent, if 5% of the drivers fail to stop at a red light, find the probability that at least 2 of the next 100 drivers fail to stop.
(b) A binary transmission channel introduces bit errors with probability 0.15. Calculate the probability that there are less than 20 errors in 100 bit transmissions.
