1.  In a certain type of automobile engine, the cylinder head is fastened to the block by 10 bolts, each of which should be torqued to 60 N · m. Assume that the torques of the bolts are independent.

a.  If each bolt is torqued correctly with probability 0.99, what is the probability that all the bolts on a cylinder head are torqued correctly?

b.   The goal is for 95% of the engines to have all their bolts torqued correctly.

c.   What must be the probability that a bolt is torqued correctly in order to reach this goal?

2.

An electronic message consists of a string of bits (0s and 1s). The message must pass through two relays before being received. At each relay the probability is 0.1 that the bit will be reversed before being relayed (i.e., a 1 will be changed to a 0, or a 0 to a 1). Find the probability that the value of a bit received at its final destination is the same as the value of the bit that was sent.

3.  Two dice are rolled. Given that two different numbers come up, what is the probability that one of the dice comes up 6?

4.   In a lot of 10 components, 2 are sampled at random for inspection. Assume that in fact exactly 2 of the 10 components in the lot are defective. Let X be the number of sampled components that are defective.

a.   Find P(X = 0).

b.   Find P(X = 1).

c.   Find P(X = 2).

d.   Find the probability mass function of X.

e.   Find the mean of X.

f.     Find the standard deviation of X.