1- In roulette there are 24 slots into which a ball may land. They are numbered 0, 00, 000, 0000 and 1 through 20. The odd numbers are red, even numbers are black, and the zeroes are green. A ball is thrown randomly into a slot.
(a) What is the probability that it is red?
(b) What is the probability it is number 7?
(c) What is the probability it is either red, or between 10 and 14, but not both?
(d) What is the probability it is not black?
2- A coin with the probability of heads being 0.7 and a fair six-sided die are used for a random experiment. The coin is first tossed. If the outcome is heads, the die is thrown twice. If the outcome is tails, the die is thrown only once. Find the probability of:
(a) Throwing a sum less than 8.
(b) Throwing a sum of which is an odd number greater than or equal to 7.
(c) Throwing a sum of 8, 10 or 12.
3- A box contains 750 silicon transistors, of which 15% are bad, and 250 germanium transistors, of which 20% are bad. If a transistor is selected at random and found to be good, what is the probability that it is germanium? If a transistor is selected at random and found to be bad, what is the probability that it is silicon?
4- A candy machine has 15 buttons to make selections. Three buttons never work, two buttons each work with a probability of 0.6, and the others work with a probability of 0.9. If the buttons are pushed at random with equal probability:
(a) Find the probability that the machine will not work on any one trial.
(b) If the machine did not work, find the probability that one of the buttons that never work was the one pushed
5- A coin with the probability of heads being 0.8 is tossed 4 times.
(a) What is the probability that the coin shows "tails" all the four times?
(b) What is the probability that 3 heads are obtained?
(c) What is the probability that at least 2 consecutive heads are obtained?
6- Assume that an old used car fails to start 10% of the time. The other 90% of the time it drives for a time period which is uniformly distributed in the interval 20 to 30 minutes. In this interval all subintervals of time of equal width between 20 to 30 minutes are equally likely.
(a) What is the probability that the car drives for at least 10 minutes given that it starts?
(b) What is the probability that the car drives for at least 10 minutes?
7- Consider two urns with the following composition: Urn 1 contains 5 red and 15 black balls, Urn 2 contains 8 red and 12 black balls. Our random experiment consists of choosing one of the two urns with equal probability and then drawing a sample (without replacement) of 3 balls. If 3 red balls are drawn, what is the probability they came from urn 1? How would the result change if the initial choice of the urn was as per the rule: Choose Urn 1 with probability 0.8 and Urn 2 with probability 0.2.
8- Assume that a class contains 20 students. If the probability of a student obtaining an "A" is 0.7, a "B" is 0.2 and a "C" is 0.1, answer the following:
(a) What is the probability that 2 students will obtain A's, 2 obtain B's, and 16 obtain C's?
(b) What is this probability if we are given that at least two students receive C's?
(c) If A has a grade value of 4, B has a grade value of 3 and C has a grade value of 1, what is the expected grade of the class? Which law of probability will you use to compute this?