Question

Statistics

EXERCISE 8. A fair pair of dies is to be cast once. What is the probability of getting a) 7, b) 11, c) 7 or 11, d) a sum divisible by 3? Answer. a) 1/6. b) 1/18. c) 2/9. d) 1/3. EXERCISE 9.One card is selected from an ordinary deck of playing cards. What is the probability of getting a) a queen, b= a jack, c= either a queen or a jack, d) a queen or a red card, e) a face card? Answer. a) 4/52. b) 4/52. c) 8/52. d) 28/52. e) 12/52.

EXERCISE 10. In a certain community, the probability that a family has a television set is 80%; a washing machine 50%; both a television set and a washing machine 45%. What is the probability that a family has either a television set or a washing machine or both? Answer. 85%.

EXERCISE 11. In a state lottery, a three-digit number is randomly selected between 000 and 999 inclusive. Find the exact probability that the number selected is less than 100 or greater than 900. Answer. 19,9%.

EXERCISE 12. A labor study involves a sample of 12 mining companies, 18 construction companies, 10 manufacturing companies and 2 wholesale companies. If a company is randomly selected from this sample group, find the probability of getting a mining or construction company. Answer. 61/366.

EXERCISE 13. Two cards are to be drawn without replacement from an ordinary deck of playing cards. What is the probability that both of the cards drawn are aces? Answer. 12/2652.

EXERCISE 14. Of all students attending a given college, 40% are males and 4% are males majoring in art. A student is to be selected at random. Given that the selected student is male, what is the probability that he is an art major? Answer. 10%

EXERCISE 15. Two cards are to be drawn without replacement from an ordinary deck of playing cards. What is the probability a) that the first card to be drawn is a queen and the second card is king? b) of drawing a combination of queen and king? c) that neither of the two cards will be queen? d) that neither of the two cards will be queen or a king? Answer. a) 16/2652. b)32/2352. c) 2256/2652. d) 1892/2652.

EXERCISE 16. Urn A contains 4 white and 3 red marbles, urn B contains 2 white and 5 red marbles. A marble is to be selected from urn A and another marble is to be selected from urn B. What is the probability that the two marbles selected are white? Answer. 8/49.

Please show your workings. I have shown the possible answer already. Thank you!

Answer 1

EXERCISE 8.

Each pair of die has numbers from 1 to 6 on them. Hence the sample space consists of 36 events which include,

= { (1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4), (4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

Here, the events are independent.

The total number of possible outcomes = N = 36

By the classical theory of probability,

If there are n equally likely outcomes for an experiment, then probability of an event A is given by,

a) Let A be the event of getting 7, then the outcomes in A are

A = {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}

Therefore, number of favorable outcomes to A = n = 6

b) Let B be the event of getting 11, then the outcomes in B are

B = {(5,6),(6,5)}

Therefore, number of favorable outcomes to B = n = 2

c) Let C be the event of getting 7 or 11.

Consider the formula,

When the events A and B are independent, .

Since the event of getting 7 and getting 11 are independent, we have,

P(getting 7 or 11) = P(getting 7) + P(getting 11) = P(A) + P(B)

d) Let D be the event of getting a sum divisible by 3.

Then the outcomes in D are

D={(1,2),(1,5),(2,1),(2,4),(3,3)(3,6),(4,2),(4,5),(5,1),(5,4),(6,3),(6,6)}

Therefore, number of favorable outcomes to D = n = 12

EXERCISE 9.

There are totally 52 cards in a deck of playing cards which consists of four suits in a deck. Each suit has 13 cards.

a) Let P be the event of getting a queen.

There is one queen card in each suit and there are four suits. Hence, there are totally four queen in a deck of playing cards.

Total number of favorable outcomes to P = n = 4

b) Let Q be the event of getting a jack.

There is one jack card in each suit and there are four suits. Hence, there are totally four jack in a deck of playing cards.

Total number of favorable outcomes to Q = n = 4

c) Let R be the event of getting a queen or a jack.

Since the event of getting a queen and the event of getting a jack are independent,

P(getting a queen or a jack) = P(getting a queen) + P(getting a jack) = P(P) + P(Q)

d) Let S be the event of getting a queen or a red card.

There are totally 26 red cards in a deck. ( 13 hearts + 13 diamonds)

There are totally 4 queen cards in a deck.

The events of getting a queen and getting a red card are not independent.

Therefore, the outcomes of the event of getting a queen and a red card is 2 ( one queen from hearts and one queen from diamonds)

P(getting a queen or a red card) = P(getting a queen) + P(getting a red card) - P(getting a queen and a red card)

e) Let T be the event of getting a face card.

In a deck of 52 cards, there are four suits in each deck. Each suit has one king, queen and jack. Face cards are king, queen and jack cards. Therefore, if there are 3 face cards in each suits, there are totally 12 face cards in a deck.

Total number of favorable outcomes to T = n = 12

EXERCISE 10.

Let X be the event that a family has a television set, then

Let Y be the event that a family has a washing machine, then

The probability that a family has both a television set and a washing machine is

The probability that a family has either a television set or a washing machine or both is

Therefore, the probability that a family has either a television set or a washing machine or both is 85%

EXERCISE 11.

We again make use of the classical theory of probability here.

Let the sample space consist of all the numbers between 000 and 999, which implies that the total number of favorable outcomes is N = 1000.

Let U be the event of selecting a number which is less than 100 or greater than 900.

The number of outcomes in U are the numbers from 000 to 099 and 991 to 999.

Hence the total number of outcomes in U = 100 + 99 =199 = n

Therefore, the exact probability that the number selected is less than 100 or greater than 900 is 19.9%