Question

1. Two cards are drawn from a well-shuffled ordinary deck of 52 cards. Find the probability that they are both aces if the first card is (a) replaced, (b) not replaced.
2. Find the probability of a 4 turning up at least once in two tosses of a fair die.

3. One bag contains 4 white balls and 2 black balls; another contains 3 white balls and 5 black balls. If one ball is drawn from each bag, find the probability that (a) both are white, (b) both are black,(c) one is white and one is black.

4. Box I contains 3 red and 2 blue marbles while Box II contains 2 red and 8 blue marbles. A fair coin is tossed. If the coin turns up heads, a marble is chosen from Box I; if it turns up tails, a marble is chosen from Box II. Find the probability that a red marble is chosen.

5. A committee of 3 members is to be formed consisting of one representative each from labor, management, and the public. If there are 3 possible representatives from labor,2 from management, and 4 from the public, determine how many different committees can be formed

6. In how many ways can 5 differently colored marbles be arranged in a row?

7. In how many ways can 10 people be seated on a bench if only 4 seats are available?

8. It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?

9. How many 4-digit numbers can be formed with the 10 digits 0,1,2,3,. . . ,9 if (a) repetitions are allowed, (b) repetitions are not allowed, (c) the last digit must be zero and repetitions are not allowed?

10. Four different mathematics books, six different physics books, and two different chemistry books are to be arranged on a shelf. How many different arrangements are possible if (a) the books in each particular subject must all stand together, (b) only the mathematics books must stand together?

11. Five red marbles, two white marbles, and three blue marbles are arranged in a row. If all the marbles of the same color are not distinguishable from each other, how many different arrangements are possible?

12. In how many ways can 7 people be seated at a round table if (a) they can sit anywhere,(b) 2 particular people must not sit next to each other?

13. In how many ways can 10 objects be split into two groups containing 4 and 6 objects, respectively?

14. In how many ways can a committee of 5 people be chosen out of 9 people?

15. Out of 5 mathematicians and 7 physicists, a committee consisting of 2 mathematicians and 3 physicists is to be formed. In how many ways can this be done if (a) any mathematician and any physicist can be included, (b) one particular physicist must be on the committee, (c) two particular mathematicians cannot be on the committee?

16. How many different salads can be made from lettuce, escarole, endive, watercress, and chicory?

17. From 7 consonants and 5 vowels,how many words can be formed consisting of 4 different consonants and 3 different vowels? The words need not have meaning.

18. In the game of poker5 cards are drawn from a pack of 52 well-shuffled cards. Find the probability that (a) 4 are aces, (b) 4 are aces and 1 is a king, (c) 3 are tens and 2 are jacks, (d) a nine, ten, jack, queen, king are obtained in any order, (e) 3 are of any one suit and 2 are of another, (f) at least 1 ace is obtained.

19. Determine the probability of three 6s in 5 tosses of a fair die.

20. A shelf has 6 mathematics books and 4 physics books. Find the probability that 3 particular mathematics books will be together.

21. A and B play 12 games of chess of which 6 are won by A,4 are won by B,and 2 end in a draw. They agree to play a tournament consisting of 3 games. Find the probability that (a) A wins all 3 games, (b) 2 games end in a draw, (c) A and B win alternately, (d) B wins at least 1 game.

22. A and B play a game in which they alternately toss a pair of dice. The one who is first to get a total of 7 wins the game. Find the probability that (a) the one who tosses first will win the game, (b) the one who tosses second will win the game.

23. A machine produces a total of 12,000 bolts a day, which are on the average 3% defective. Find the probability that out of 600 bolts chosen at random, 12 will be defective.

24. The probabilities that a husband and wife will be alive 20 years from now are given by 0.8 and 0.9, respectively. Find the probability that in 20 years (a) both, (b) neither, (c) at least one, will be alive.

ok, I'll update this to 3 to 4 questions. Thanks!

Answer 1

Solution

Back-up Theory

Probability of an event E, denoted by P(E) = n/N ………………...................………………………...………(1)

where

n = n(E) = Number of outcomes/cases/possibilities favourable to the event E and

N = n(S) = Total number all possible outcomes/cases/possibilities.

Number of ways of selecting r things out of n things is given by ⁿC_r = (n!)/{(r!)(n - r)!}….........…(2)

Number of ways of selecting r things out of n things when the same thing can be selected

any number of times (i.e., with replacement) is given by n^r........................................................ (2a)

Values of ⁿC_r can be directly obtained using Excel Function: Math & Trig COMBIN…...........…. (2b)

Now to work out the solution,

Q1

Part (a)

Vide (2a),

Two cards can be drawn from a well-shuffled ordinary deck of 52 cards in 52²

= 2704 ways. So, vide (1), N = 2704 ........................................................................................... (3)

There are 4 aces in a deck of 52 cards. So, number of selections of 2 aces = 4² = 16

So, vide (1), n = 16 ..................................................................................................................... (3a)

Hence, vide (1), (3) and (3a)the required probability = 16/2704 = 0.0059 Answer 1

Part (b)

Vide (2),

Two cards can be drawn from a well-shuffled ordinary deck of 52 cards in ⁿC_r

= 1326 ways. So, vide (1), N = 1326 ........................................................................................... (4)

There are 4 aces in a deck of 52 cards. So, number of selections of 2 aces = ⁴C₂= 6

So, vide (1), n = 6 ..................................................................................................................... (4a)

Hence, vide (1), (4) and (4a)the required probability = 6/1326 = 0.0045 Answer 2

Q2

Probability of a 4 turning up in one toss of a fair die = 1/6 and hence probability of

a 4 not turning up in one toss of a fair die = 5/6.

So, probability of a 4 turning up in neither of two tosses of a fair die = (5/6)² = 0.6944 .

Now,

Probability of a 4 turning up at least once in two tosses of a fair die

= 1 - probability of a 4 turning up in neither of two tosses of a fair die

= 1 - 0.6944

= 0.3056 Answer 3

Q3

Part (a)

If one ball is drawn from a bag that contains 4 white balls and 2 black balls,the probability that it is white = 4/6 = 2/3. If one ball is drawn from another bag that contains 3 white balls and 5 black balls,the probability that it is white = 3/8 .

Hence the probability that both are white = (2/3)(3/8) = ¼. Answer 4

Part (b)

Similarly, the probability that both are black = (1/3)(5/8) = 5/24 Answer 5

Part (c)

Similarly, the probability that one is black and the other is white

= (1/3)(3/8) + (2/3)(5/8) = 13/24 Answer 6

DONE