1) 10 points
A string of Xmas tree lights contains 10 bulbs, all of which must be good in order for the string of lights to light up. If the probability that any individual bulb is bad is 0.1, what is the probability that the string of lights will not light up?
2) 10 points
The probability of a salesman making a sale on a single call is 1/6. What is the probability that he will make at least one sale in the next five calls? What is the probability that he will make four or more sales in five calls?
3) 10 points
Consider an ordinary deck of playing cards, in which the 52 cards have been ordered randomly.
a) If four cards are dealt from the top of the deck, what is the probability that the 1st and the 4th card will be a king, while the 2nd and 3rd cards are not?
b) What is the probability that at least two kings are dealt in the first four cards?
4) 10 points
Consider filling a jar with colored balls using the following method: A fair coin is tossed. If the coin comes up heads a red ball is placed in the jar, and if the coin comes up tails a blue ball is placed in the jar. This is done four times, so that there are four balls in the jar. The jar is now brought to someone who picks two balls out of the jar.
a) What is the probability that both balls picked from the jar are red?
[Hint: Consider the probability both picked balls are red, if all balls in the jar are red; if 3 balls in the jar are red; etc. Then, what is the probability all balls in the jar were indeed red; what is probability 3 balls were the jar are red; etc. Then put all this information together.]
b) If both balls picked from the jar do come out red, what is the probability that the jar contained two red balls and two blue balls?
[Hint: think of how we did part (c) in of the flu epidemic problem - example 6 in the text, or how we did the two lying brothers problem.]