I

I. CONCEPT OF PROBABILITY

1. From the integers from 2 to 40 inclusive, one integer is selected at random. What is the probability that it is

(a) a perfect square, (b) a prime, (c) a multiple of 9,

(d) a perfect square or a prime, (e) a perfect square or a multiple of 9.

2. Comment on each of the following statements:

(a) "In a tennis championship, there are 32 entrants. The probability that each entrant will win is 1/32"

(b) "If 2 unbiased coins are tossed, since there are 3 possible occurrences - 2 heads, a head and a tail, 2 tails - then the probability of 2 tails is 1/3".

3. (i) A girl has 5 pairs of shoes, each pair being a different style. If she selects 2 shoes at random, what is the probability they form a matching pair ?

(ii) Three unbiased coins are tossed, what is the probability of obtaining (a) 3 tails , (b) one tail and two heads.

4. (i) Two dice, each numbered 1 to 6, are tossed. Find the probability that

(a) both show a "4" (b) neither shows a "4"

(ii) If the dice are thrown n times, find the probability of obtaining

(a) no double "4" (b) at least one double "4" in the n throws.

Hence determine the least value of n for which the probability of at least one double 4 is greater than 1/2.

5. (i) An urn contains 7 red and 4 blue balls. Two balls are drawn without replacement. What is the probability that the balls are

(a) both red (b) a red and a blue (c) both the same colour.

(ii) Find the corresponding probabilities if the balls are replaced after each draw.

6. (i) By listing all possibilities, find the number of committees of 3 possible from 5 persons A, B, C, D, and E. If all persons are equally likely to be elected, what is the probability that A is not on the committee.

(ii) At a carnival, the probability that a boy will win the 100 m event is 1/10, and the probability he wins the 200 m event is 1/8. Find the probability that he wins one of these events.

7. In a group A, there are 7 boys and 3 girls, whilst in another group B, there are 2 boys and 8 girls.

(i) One person is selected at random from each group. Find the probability that (a) both are boys (b) one is a boy and one is a girl.

(ii) If two persons are selected from either of the original groups A and B, find the probability that (a) both are boys (b) one is a boy and one is a girl.

8. (i) The probability that a certain variety of plant produces a yellow flower is 1/5. If two plants are selected at random, what is the probability that:

(a) both produces yellow flowers

(b) only one produces a yellow flower

(d) at least one produces a yellow flower.

(ii) If n of these plants are selected, what is the probability that there is at least one yellow flower. Hence, determine the least number of plants which need to be chosen so that the probability of at least one yellow flower is greater than 0.99.

9. (i) The probability that a biased coin falls heads is 1/4. Find the probability of at least one head in (a) 2 throws, (b) 3 throws , of the coin.

(ii) Determine the least number of times the coin must be thrown so that the probability of at least one head occurring is greater than 0.999.

10. One of 5 pens is faulty. They are tested one at a time, what is the probability that the faulty pen is :

(a) 1th first one tested (b) the 2nd one testedt

(b) the rd one tested (d) the last one tested

11. (i) How many possibilities are there if (a) 2 (b) 3 and (c) 4 ordinary dice (numbered 1,2,3,4,5,6) are thrown ?

(ii) Find the probability of total score of 7 with (a) 2 dice (b) 3 dice (c) 4 dice. hint: carefully list these cases giving a total score of 7 with 2,3,4 dice. Do not write out the sample space).

12. (i) There are two lines of defence against an attacking aircraft, the first is a missile and the second an anti-aircraft gun. The respective probabilities that the aircraft being hit by the missile and the gun are 2/9 and 1/8, respectively. What is the probability of hitting the aircraft ?

(ii) Two girls have their birthdays in the same week. It is known that one of the girls was born on a Saturday, write down the sample space and hence determine the probability that both were born a Saturday.

13. (i) From an ordinary deck of cards, only 12 picture cards are retained. They are shuffled (mixed) and a man draws 2 cards at random. He then announce one of these cards is the King or Spades. What is the probability that he holds two Kings? (hint: list the possible sample space carefully).

(ii) The 2 cards are replaced and the cards re-shuffled. He then draws 2 cards, and announces that he holds at least one King. Find the probability that he holds two Kings in his hand.

14. An a small raffle, there are only 2 prizes, and 30 tickets are sold. A boy has 3 tickets in the raffle. Find the probability that he wins (a) first prize, (b) both prize, (c) only the 2nd prize, and (d) a prize.

15. In a raffle in which 200 tickets are sold, there are 3 prizes - first prize of $100, second prize of $50, third prize of $30.

(i) A girl has one ticket in the raffle. What is the probability that she wins (a) a prize, (b) at least $50.

(ii) A man has 2 tickets in this raffle. WHat is the probability that he wins (a) a prize, (b) both prizes.

16. In a bag of balloons, the ratio of red balloons to other colours is 1:5, if 3 balloons are selected from the bag at random, what is the probability that (a) at least one balloon is red, (b) exactly one balloon is red.

17. If the probabilities that three students A, B and C may pass the HSC are 4/7, 3/5, and 2/3, respectively. What is the probability that

(a) all 3 pass (b) all 3 fail

(e) at least two pass (f) at least one passes.

18*. A point is moving randomly on a line of length L. What is the probability that the ratio of the shorter to the longer segment is less than 1/3 ?

19*. A point is moving randomly inside an equilateral triangle whose side length is 3 cm. What is the probability that its distance to any corner is greater than 1 ?

20*. Two persons agree to meet at between 2 PM to 4 PM, but each of them will wait 30 minutes for the late comer. What is the probability that they will meet ?

ANSWERS: 1.(a) 5/39 (b) 4/13 (c) 4/39 (d) 17/39 (e) 3/13.

3.(i) 1/9 (ii) (a) 1/8 (b) 3/8 4.(i)(a) 1/36 (b) 25/36 (c) 11/36 (d) 1/9 (ii) (a) (35/36)n

(b) 1-(35/36)n; min n is 25 5. (i) (a) 21/55 (b) 28/55 (c) 27/55 (ii) (a) 41/121 (b) 56/121

(ii) (7+9)/(80=1/5. 7.(i) (a) 7/50 (b) 31/50 (ii) (a) 11/45 (b) 37/90.

8.(i) (a) 1/25 (b) 8/25 (c) 16/25 (d) 9/25 (ii) 1-(4/5)n least n is 21.

9.(i) (a) 1-3/4)2 = 7/16 (b) 1-(3/4)3 =37/64 (c) 1-(3/4)n >0.999 when (4/3)n >1000; least

n is 25. 10.(a) 1/5 (b) 1/5 (c) 1/5 (d) 1/5. 11. (a) 36 (b) 216 (c) 1296

(ii) (a) 1/6 (b) 5/72 (c) 5/324. 12.(i) 23/72 (ii) 1/13. 13.(i) 3/11 (ii) 3/19. 14.(a) 1/10

(b) 1/145 (c) 27/290 (d) 28/145. 15. (i) (a) 3/200 (b) 1/100 (ii) (a) 297/9950 (b) 1/19900

16.(a) 91/216 (b) 25/72. 17.(a) 8/35 (b) 2/35 (c) 46/105 (d) 29/105 (e) 2/5 (f) 33/35.

18.1/2 19. 20. 7/16.