Probability (Ex – 16.1)
In each of the following Exercises 1 to 7, describe the sample space for the indicated experiment.
Question 1.
A coin is tossed three times.
Solution:
When one coin is tossed three times, The sample space of the experiment is given by S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}.
Question 2.
A die is thrown two times.
Solution:
When a die is thrown two times. The sample space S for this experiment is given by
S = {(1, 1), (1, 2),(1, 6), (2, 1), (2, 2), … (2, 6), …, (6, 1),…, (6, 6)}.
Question 3.
A coin is tossed four times.
Solution:
When a coin is tossed four times. The sample space S for this experiment is given by
S = {HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT}.
Question 4.
A coin is tossed and a die is thrown.
Solution:
When a coin is tossed and a die is thrown. The sample space S for the experiment is given by
S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}.
Question 5.
A coin is tossed and then a die is rolled only in case a head is shown on the coin.
Solution:
When a coin is tossed and a die is rolled only in case if head is shown on the coin. The sample space S for the experiment is given by
S = {H1, H2, H3, H4, H5, H6, T}.
Question 6.
2 boys and 2 girls are in Room X, and 1 boy and 3 girls in Room Y. Specify the sample space for the experiment in which a room is selected and then a person.
Solution:
Let B1, B2 and G1, G2 are the boys and girls respectively in room X, B3 and G3, G4, G5 are the boy and girls respectively in room Y. The sample space S for the experiment is given by
S = {XB1, XB2, XG1, XG2, YB3, YG3, YG4, YG5).
Question 7.
One die of red color, one of white color and one of blue color are placed in a bag. One die is selected at random and rolled, its color and the number on its uppermost face is noted. Describe the sample space.
Solution:
Let R, W and B denote the red, white and blue dice respectively. The sample space S, for this experiment is given by
S = {R1, R2, R3, R4, R5, R6, W1, W2, W3, W4, W5, W6, B1, B2, B3, B4, B5, B6}.
Question 8.
An experiment consists of recording boy-girl composition of families with 2 children.
(i) What is the sample space if we are interested in knowing whether it is a boy or girl in the order of their births?
(ii) What is the sample space if we are interested in the number of girls in the family?
Solution:
(i) The sample space S, in knowing whether it is a boy or a girl in the order of their births in composition of families with two children is S = {BB, BG, GB, GG}.
(ii) The sample space S, in knowing the number of girls in a family is S = {0,1, 2}.
Question 9.
A box contains 1 red and 3 identical white balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.
Solution:
There are 1 red and 3 identical white balls in a box. The sample space S for this experiment is given by S = {RW, WR, WW}.
Question 10.
An experiment consists of tossing a coin and then throwing it second time if a head occurs. If a tail occurs on the first toss, then a die is rolled once. Find the sample space.
Solution:
The sample space S, for tossing a coin and then tossing it second time if a head occurs; if a tail occurs on the first toss, the die is tossed once is given by S = {HH, HT, T1, T2, T3, T4, T5, T6}.
Question 11.
Suppose 3 bulbs are selected at random from a lot. Each bulb is tested and classified as defective (D) or non-defective (N). Write is the sample space of this experiment?
Solution:
The sample space S for selecting three bulbs at random from a lot is
S = {DDD, DDN, DND, NDD, DNN, NDN, NND, NNN}.
Question 12.
A coin is tossed. If the out come is a head, a die is thrown. If the die shows up an even number, the die is thrown again. What is the sample space for the experiment?
Solution:
An experiment consists of tossing a coin. If the result is a head, a die is thrown. If the die shows up an even number, the die is thrown again. The sample space S for this experiment is
S = {T, H1, H3, H5, H21, H22, H23, H24, H25, H26, H41, H42, H43, H44, H45, H46, H61, H62, H63, H64, H65, H66}.
Question 13.
The numbers 1,2,3 and 4 are written separately on four slips of paper. The slips are put in a box and mixed thoroughly. A person draws two slips from the box, one after the other, without replacement. Describe the sample space for the experiment.
Solution:
Four slips marked as 1,2,3 and 4 are put in a box. Two slips are drawn from it one after the other without replacement. The sample space S, for the experiment is S = {(1, 2), (1, 3), (1, 4), (2,1), (2, 3), (2, 4), (3,1), (3, 2), (3, 4), (4,1), (4, 2), (4, 3)}.
Question 14.
An experiment consists of rolling a die and then tossing a coin once if the number on the die is even. If the number on the die is odd, the coin is tossed twice. Write the sample space for this experiment.
Solution:
An experiment consists of rolling a die and then tossing a coin once if the number on the die is even. If the number is odd, the coin is tossed twice. The sample space S for this experiment is given by S = {1HH, 1TH, 1HT, 1TT, 2H, 2T, 3HH, 3HT, 3TH, 3TT, 4H, 4T, 5HH, 5HT, 5TH, 5TT, 6H, 6T}.
Question 15.
A coin is tossed. If it shows a tail, we draw a ball from a box which contains 2 red and 3 black balls. If it shows head, we throw a die. Find the sample space for this experiment.
Solution:
An experiment consists of tossing a coin. If it shows a tail, a ball is drawn from a box which contains 2 red and 3 black balls. If it shows head, a die is thrown. Then the sample S for this experiment is given by
S = {TR1, TR2, TB1, TB2, TB3, H1, H2, H3, H4, H5, H6)
Question 16.
A die is thrown repeatedly until a six comes up. What is the sample space for this experiment?
Solution:
An experiment consists of rolling a die.
∴ Sample space = {6, (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (1, 1, 6), (1, 2, 6), (1, 3, 6),…, (1, 5, 6), (2, 1, 6), (2, 2, 6),…, (2, 5, 6),…, (5, 1, 6), (5, 2, 6),…}.
