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Four probability problems

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1. The Monty Hall problem is as follows: A contestant on a game show is given the opportunity to select one of three curtains on stage. One curtain hides a valuable prize and the other two hide worthless prizes. After the contestants selection, Monty, who knows what is behind each curtain, draws back one of the remaining curtains to show a worthless prize and offers the contestant the chance to exchange the selected curtain for the one remaining. Should the contestant keep the curtain originally selected or exchange it for the remaining? Verfiy your answer using probabilities.

2. You have a standard deck of cards and draw five cards at random. What is the probability you draw three kings?

3. Supppose you have two urns, each containing red and green balls. THe first urn contains 7 red and 3 green balls and the second urn contains 4 red and 5 green balls. Suppose now that you randomly select a ball from the first urn and place it (unseen) into the second urn. You then randomly select a ball from the second urn.
a. What is the probability that the ball you selected from the second urn is green?
b. What is the probability that the ball you selected from the second urn is green, given that the ball selected from the first urn was red?

4. A sales representative for a tire manufacture claims that the companies steel-belted radials get at least 35,000 miles. A tire dealer decides to check this claim by testing eight of the tires. If 75% or more of the eight tires he tests get at least 35,000 miles, he will purchase tires from the representative. IF, in fact, 90% of the steel-belted radials get at least 35,000 miles, what is the probability that the tire dealer purchases tires from the sales representative?