NCERT Solutions for Exercise 16.1 Class 11 Maths Chapter 16 - Probability

Probability Class 11 Chapter 16 Exercise 16.1

Question:1 Describe the sample space for the indicated experiment.

A coin is tossed three times.

Answer:

Let H denote Heads and T denote Tails.

For each toss, there are two possible outcomes = H or T

The required sample space is:

S = {HHH, HHT, HTH, THH, TTH, HTT, THT, TTT}

Question:2. Describe the sample space for the indicated experiment.

A die is thrown two times.

Answer:

When a die is thrown, the possible outcomes are = {1, 2, 3 ,4 ,5 ,6}

The required sample space is:

S = { = 1,2,3,4,5,6}

or S = {(1,1), (1,2), (1,3), ..., (1,6), (2,1), (2,2), ..., (2,6), ..., (6, 1), (6, 2), ..., (6,6)}

Question:3 Describe the sample space for the indicated experiment.

A coin is tossed four times.

Answer:

Let H denote Heads and T denote Tails.

For each toss, there are two possible outcomes = H or T

The required sample space is:

S = {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT}

Question:4 Describe the sample space for the indicated experiment.

A coin is tossed and a die is thrown.

Answer:

Let H denote Heads and T denote Tails.

For each toss, there are two possible outcomes = H or T

And,

When a die is thrown, the possible outcomes are = {1, 2, 3 ,4 ,5 ,6}

The required sample space is:

S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

Question:5 Describe the sample space for the indicated experiment.

A coin is tossed and then a die is rolled only in case a head is shown on the coin.

Answer:

Let H denote Heads and T denote Tails.

For each toss, there are two possible outcomes = H or T

For H, when a die is thrown, the possible outcomes are = {1, 2, 3 ,4 ,5 ,6}

The required sample space is:

S = {H1, H2, H3, H4, H5, H6, T}

Question:6 Describe the sample space for the indicated experiment.

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.

Answer:

Let X denote the event Room X is selected, Y denote the event Room Y is selected.

B1, B2 denote the event a boy is selected and G1, G2 denote the event a girl is selected from room X.

B3 denotes the event a boy is selected and G3, G4, G5 denote the event a girl is selected from Room Y.

The required sample space is:

S = {XB1 , XB2 , XG1 , XG2 , YB3 , YG3 , YG4 , YG5 }

Question:7. Describe the sample space for the indicated experiment.

One die of red colour, one of white colour and one of blue colour are placed in a bag. One die is selected at random and rolled, its colour and the number on its uppermost face is noted. Describe the sample space.

Answer:

Let, R denote the event the red die comes out,

W denote the event the white die comes out,

B denote the event the Blue die is chosen

When a die is thrown, the possible outcomes are = {1, 2, 3 ,4 ,5 ,6}

The required sample space is:

S = {R1, R2, R3, R4, R5, R6, W1, W2, W3, W4, W5, W6, B1, B2, B3, B4, B5, B6}

Question:8(i) An experiment consists of recording boy–girl composition of families with 2 children.

What is the sample space if we are interested in knowing whether it is a boy or girl in the order of their births?

Answer:

Let B denote the event a boy is born,

G denote the event a girl is born

The required sample space with a boy or girl in the order of their births is:

S = {BB, BG, GB, GG}

Question:8(ii). An experiment consists of recording boy–girl composition of families with 2 children.

What is the sample space if we are interested in the number of girls in the family?

Answer:

(ii) In a family with two child, there can be only three possible cases:

no girl child, 1 girl child or 2 girl child

The required sample is:

S = {0, 1, 2}

Question:9 A box contains red and identical white balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.

Answer:

Given, Number of red balls =1

Number of white balls = 3

Let R denote the event that the red ball is drawn.

And W denotes the event that a white ball is drawn.

Since two balls are drawn at random in succession without replacement,

if the first ball is red, the second ball will be white. And if the first ball is white, second can be either of red and white

The required sample space is:

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.

Answer:

Let H denote the event that Head occurs and T denote the event that Tail occurs.

For T in first toss, the possible outcomes when a die is thrown = {1, 2, 3 ,4 ,5 ,6}

The required sample space is :

S = {HH, HT, T1, T2, T3, T4, T5, T6}

Question:11. Suppose bulbs are selected at random from a lot. Each bulb is tested and classified as defective (D) or non – defective(N). Write the sample space of this experiment.

Answer:

Let D denote the event the bulb is defective and N denote the event the bulb is non-defective
The required sample space 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?

Answer:

Possible outcomes when a coin is tossed = {H,T}

Possible outcomes when a die is thrown = {1,2,3,4,5,6}

When T occurs, experiment is finished. S1 = {T}

When H occurs, a die is thrown.

If the outcome is odd ({1,3,5}), S2 = {H1, H3, H5}

If the outcome is even({2,4,6}), the die is thrown again.,

S3 = {H21, H22, H23, H24, H25, H26, H41, H42, H43, H44, H45, H46, H61, H62, H63, H64, H65, H66}

The required sample space 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 and are written separatly 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.

Answer:

Given, two slips are drawn from the box, one after the other, without replacement.

Let 1, 2, 3, 4 denote the event that 1, 2, 3, 4 numbered slip is drawn respectively.

When two slips are drawn without replacement, the first event has 4 possible outcomes and the second event has 3 possible outcomes

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.

Answer:

Possible outcomes when a die is thrown = {1,2,3,4,5,6}

Possible outcomes when a coin is tossed = {H,T}

If the number on the die is even {2,4,6}, the coin is tossed once.

S1 = {2H, 2T, 4H, 4T, 6H, 6T}

If the number on the die is odd {1,3,5}, the coin is tossed twice.

S2 = {1HH, 1HT, 1TH, 1TT, 3HH, 3HT, 3TH, 3TT, 5HH, 5HT, 5TH, 5TT}

The required sample space is:

S = {1HH, 1HT, 1TH, 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 red and black balls. If it shows head, we throw a die. Find the sample space for this experiment.

Answer:

Possible outcomes when a coin is tossed = {H,T}

Possible outcomes when a die is thrown = {1,2,3,4,5,6}

Let R1 and R2 denote the event that a red ball is drawn

and B1, B2, B3 denote the event that a blue ball is drawn

If H occurs, a die is thrown.

S1 = {H1, H2, H3, H4, H5, H6}

If T occurs, a ball from a box which contains red and black balls is drawn.

S2 = {TR1 , TR2 , TB1 , TB2 , TB3}

The required sample space is:

S = {TR1 , TR2 , TB1 , TB2 , TB3 , H1, H2, H3, H4, H5, H6}

Question:16 A die is thrown repeatedly untill a six comes up. What is the sample space for this experiment?

Answer:

Given, a die is thrown repeatedly untill a six comes up.

Possible outcomes when a die is thrown = {1,2,3,4,5,6}

In the experiment 6 may come up on the first throw, or the 2nd throw, or the 3rd throw and so on till 6 is obtained.

The required sample space is:

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