Stats Probability problems

1. A committee consists of five Chicanos, two Asians, three African
Americans, and two Caucasions.

A subcommittee of four is chosen at random. What is the probability that
all the ethnic groups are represented on the subcommittee?

Answer the question for part (a) if a subcommittee of five is chosen.

2. The game of Mastermind starts in the following way: One player
selects four pegs, each peg having six possible colors, and places them
in a line. The second player then tries to guess the sequence of colors.
What is the probability of guessing correctly?

3. An urn contains tree red and two white balls. A ball is drawn, and
then it and another ball of the same color are placed back in the urn.
Finally, a second ball is drawn.

What is the probability that the second ball drawn is white?

If the second ball is white, what is the probability that the first ball
drawn was red?

4. A factory runs three shifts. In a given day, 1% of the items produced
by the first shift are defective, 2% of the second shift’s items are
defective, and 5% of the third shift’s items are defective. If the
shifts all have the same productivity, what percentage of the items
produced in a day are defective? If an item is defective, what is the
probability that it was produced by the third shift?

5. Here is a simple model of a queue. The queue runs in discrete time (t
= 0,1,2…), and at each unit of time the first person in the queue is
served with probability p, and independently, a new person arrives with
probability q. At time t = 0, there is one person in the queue. Find the
probabilities that there are 0,1,2,3 people in line at time = 2.