UNIT-8: PROBABILITY THEORY

VENN DIAGRAM
Q. 1. In a survey carried out in a school snack shop, the following results were obtained. Of 100 boys
questioned, 78 liked sweets, 74 ice-cream, 53 cake, 57 liked both sweets and ice-cream. 46 liked both sweets
and cake while only 31 boys liked all three. If all the boys interviewed liked at least one item, draw a Venn
diagram to illustrate the results. How many boys liked both ice-cream and cake?
Q. 2. What is the probability of throwing a "six" with a single die?
Q.3. A box contains 7 red, 5 white, and 4 black balls. What is the probability of your drawing at random one
red ball? One black ball?
Q. 4. What is the probability of making a 7 in one throw of a pair of dice?
Solution: There are 6 × 6 = 36 ways that two dice can be thrown, as shown in the accompanying figure.
Q. 5. Suppose that we have a bag containing two red balls, three white balls, and six blue balls. What is the
probability of obtaining a red or a white ball on one withdrawal?
Q. 6. A bag contains 4 white balls, 6 black balls, 3 red balls, and 8 green balls. If one ball is drawn from the
bag, find the probability that it will be either white or green.
Q. 7. A coin is tossed nine times. What is the total number of possible outcomes of the nine-toss experiment?
How many elements are in the subset "6 heads and 3 tails"? What is the probability of getting exactly 6
heads and 3 tails in nine tosses of this unbiased coin?
Q. 8. In a single throw of a pair of dice, find the probability of obtaining a total of 4 or less.
Q. 9. A card is drawn at random from a deck of cards. Find the probability that at least one of the following
three events will occur:
Event A: a heart is drawn.
Event B: a card which is not a face card is drawn.
Event C: the number of spots (if any) on the drawn card is divisible by 3.
Q. 10. What is the probability of getting a 5 on each of two successive rolls of a balanced die?
CONDITIONAL PROBABILITY
Q. 11. Find the probability that a face card is drawn on the first draw and an ace on the second in two
consecutive draws, without replacement, from a standard deck of cards.
Q. 12. A survey was made of 100 customers in a department store. Sixty of the 100 indicated they visited the
store because of a newspaper advertisement. The remainder had not seen the ad. A total of 40 customers
made purchases; of these customers, 30 had seen the ad. What is the probability that a person who did not
see the ad made a purchase? What is the probability that a person who saw the ad made a purchase?
Q. 13. A coin is tossed 3 times and 2 heads and 1 tail fall. What is the probability that the first toss was
heads?

Q. 14. A coin is tossed 3 times. Find the probability that all 3 are heads,
(a) if it is known that the first is heads,
(b) if it is known that the first 2 are heads,
(c) if it is known that 2 of them are heads.

Q. 15. A hand of five cards is to be dealt at random and without replacement from an ordinary deck of 52
playing cards. Find the conditional probability of an all spade hand given that there will be at least 4 spades
in the hand.

Q. 16. A bag contains two balls, each of which may be either red or white. A ball is drawn at random and
found to be red. What is the probability that the other ball is red?
Q. 17. A bowl contains eight chips. Three of the chips are red and the remaining five are blue. If two chips
are drawn successively, at random and without replacement, what is the probability that the first chip drawn
is red and the second drawn is blue?

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Q. 18. If 4 cards are drawn at random and without replacement from a deck of 52 playing cards, what is the
chance of drawing the 4 aces as the first 4 cards?
Q. 19. Four cards are to be dealt successively, at random and without replacement, from an ordinary deck of
playing cards. Find the probability of receiving a spade, a heart, a diamond, and a club, in that order.
Q. 20. Your company uses a pre-employment test to screen applicants for the job of repairman. The test is
passed by 60% of the applicants. Among those who pass the test 80% complete training successfully. In an
experiment, a random sample of applicants who do not pass the test is also employed. Training is
successfully completed by only 50% of this group. If no pre-employment test is used, what percentage of
applicants would you expect to complete training successfully?
Q. 21. A bag contains 1 white ball and 2 red balls. A ball is drawn at random. If the ball is white then it is put
back in the bag along with another white ball. If the ball is red then it is put back in the bag with two extra
red balls. Find the probability that the second ball drawn is red. If the second ball drawn is red, what is the
probability that the first ball drawn was red?
Note that in order to compute these probabilities we needed to know the number of red and white balls in the
bag at the beginning.

BAYES’ THEOREM
Q. 22. Twenty percent of the employees of a company are college graduates. Of these, 75% are in
supervisory position. Of those who did not attend college, 20% are in supervisory positions. What is the
probability that a randomly selected supervisor is a college graduate?
Q. 23. In a factory four machines produce the same product. Machine A produces 10% of the output,
machine B, 20%, machine C, 30%, and machine D, 40%. The proportion of defective items produced by
these follows: Machine A: .001; Machine B: .0005; Machine C: .005; Machine D: .002. An item selected at
random is found to be defective. What is the probability that the item was produced by A? by B? by C? by
D?
Q. 24. In the St. Petersburg Community College, 30% of the men and 20% of the women are studying
mathematics. Further, 45% of the students are women. If a student selected at random is studying
mathematics, what is the probability that the student is a woman?

RANDOM SAMPLING

Q. 25. If 4 different balls are placed at random in 3 different cells, find the probability that no cell is empty.
Assume that there is ample room in each cell for all 4 balls.
Q. 26. A box contains 4 black marbles, 3 red marbles, and 2 white marbles. What is the probability that a
black marble, then a red marble, then a white marble is drawn without replacement?
Q. 27. Find the probability of drawing three consecutive face cards on three consecutive draws (with
replacement) from a deck of cards.
Let: Event A: face card on first draw,
Event B: face card on second draw, and
Event C: face card on third draw.