Solved: A pressure control apparatus contains 4 valves.

Three screws are drawn at random from a lot of 100 screws. 10 of which are defective. Find the probability that the screws drawn will be nondefective in drawing (a) with replacement. (b) without replacement.

In Prob. I find the probability of E: At least I defective (i) directly. (ii) by using complements; in both cases (a) and (b).

If we inspect paper by drawing 5 sheets without replacement from every batch of 500. what is the probability of getting 5 clean sheets although 2% of the sheets contain spots') First guess.

Under what conditions will it make practically no difference whether we sample with or witholll replacement? Give numeric examples.

If you need a right -handed screw from a box containing 20 right-handed and 5 left-handed screws. what is the probability that you get at least one right-handed screw in drawing 2 screws with replacement?

If in Prob. 5 you draw without replacement. does the probability decrease or increase? First think, then calculate.

What gives the greater probability of hitting some target at least once: (a) hitting in a shot with probability 112 and firing I shot. or (b) hitting in a shot with probability 1I4 and firing 2 shots? First guess. Then calculate.

Suppose that we draw cards repeatedly and with replacement from a file of 100 cards. 50 of which refer to male and 50 to female persons. What is the probability of obtaining the second "female" card before the third "male" card?

What is the complementary event of the event considered in Prob. 8'1 Calculate its probability and use it to check your result in Prob. 8.

In rolling two fair dice. what is the probability of obtaining a sum greater than 4 but not exceeding 7'1

In roIling two fair dice. what is the probability of obtaining equal numbers or numbers with an even product?

Solve Prob. II by considering complements.

A motor drives an eiectIic generator. During a 30-day period. the motor needs repair with probability 8%- and the generator needs repair with probability 49'r. What is the probability that during a given period. the entire apparatus (consisting of a motor and a generator) will need repair?

If a circuit contains 3 automatic switches and we want that. with a probability of 959'c. during a given time interval they are all working. what probability of failure per time interval can we admit for a single switch?

If a certain kind of tire has a life exceeding 25 000 miles with probability 0.95. what is the probability that a set of 4 of these tires on a car will last longer than 25000 miles?

In Prob. 15. what is the probability that at least one of the tires will not last for 25 000 miles?

A pressure control apparatus contains 4 valves. The apparatus will not work unless all valves are operative. If the probability of failure of each valve during some interval of time is 0.03. what is the corresponding probability of failure of the apparatus?

Show that if B is a subset of A, then P(B) peA).

Extending Theorem 4. show that P(A n B n C) = P(A)P(BIA)P(CiA n B).

You may wonder whether in (16) the last relation follows from the others. but the answer is no. To see this. imagine that a chip is drawn from a box containing 4 chips numbered 000,01 I. 101, 110. and let A. B. C be the events that the first. second. and third digit. respectively. on the drawn chip is 1. Show that then the first three formulas in (16) hold but the last one does not hold.