**In each of the following Questions from 1 to 7, describe the sample space for the indicated experiment.**

**Question 1. A coin is tossed three times**.

**Solution: **

Suppose if you have a coin in your pocket and you want to play head and tails with the coin. The condition of the game is you should only toss the coin for three times so, when you flip a coin there is an equal chance of getting both head(H) and tail(T) right?

So, there are 2 possibilities and you flipped the coin 3 times which

is 2^3 = 8.The above discussion says there are 8 possible outcomes when you flipped a coin

3 times, and they are {HHH, THH, HTH, HHT, TTT, HTT, THT, TTH}

**Question 2. A die is thrown two times.**

**Solution: **

When you roll a die there are 6 possible outcomes

So, if you roll the same die for 2 times there are 6*6 = 36 possible outcomes and they are:

{(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)}

**Question 3. A coin is tossed four times.**

**Solution: **

There are 2 possible outcomes head(H) and tail(T) when you flip a coin.

If you flip the coin for 4 times there will are 2*2*2*2 = 2^4 = 16 possible outcomes, and they are:{HHHH, THHH, HTHH, HHTH, HHHT, TTTT, HTTT, THTT, TTHT, TTTH, TTHH, HHTT,

THTH, HTHT, THHT, HTTH}

**Question 4. A coin is tossed and a die is thrown.**

**Solution: **

When you flip a coin there are 2 possible outcomes head(H) and tail(T) when you roll a die there are 6 possible outcomes 1,2,3,4,5,6

So, the possible outcomes will be 2*6 = 12, and they are:

{(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}

**Question 5. A coin is tossed and then a die is rolled only in case a head is shown on the coin.**

**Solution: **

There are 2 possible outcomes head(H) and tail(T) when you flip a coin.

When you roll a die there are 6 possible outcomes 1, 2, 3, 4, 5, 6

So, when you get a head the possible outcomes of a die will be 6.And there is also a chance of getting a tail i.e. 1 so, the possible outcomes will be 6+1 = 7, and they are:

{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 us take boys as B1, B2 and girls as G1, G2 in room X in the same way let us take boys as B3 and girls as G3, G4, G5 in room Y.

If room X is selected the possible outcome will be {(X,B1), (X,B2), (X,G1), (X,G2)}

If room Y is selected the possible outcome will be {(Y,B3), (Y,G3), (Y,G4), (Y,G5)}The total sample space will be: {(X,B1), (X,B2), (X,G1), (X,G2), (Y,B3), (Y,G3), (Y,G4), (Y,G5)}

**Question 7. 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.**

**Solution: **

If we roll the die there are 6 possible outcomes 1, 2, 3, 4, 5, 6 and if we roll the same die for three times the possible outcomes will be 6*3 = 18.

Assume that red colour, white colour and blue colour are denoted as R, W, B respectively.

Then the all 18 possible outcomes(sample space) will be:

{(R,1),(R,2),(R,3),(R,4),(R,5),(R,6),(W,1),(W,2),(W,3),(W,4),(W,5),(W,6) (B,1),(B,2),(B,3),(B,4),(B,5),(B,6)}

**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: **

Assume boy as B and girl as G

i)If it is a boy or girl we have the sample space as {GG, BB, GB, BG}ii)If it is a girl and if there are 2 girl child’s in the family the sample space is {GG} and if there is one girl child in the family the sample space will be (GB, BG} so, in total we have a sample space as {GG, GB, BG}

**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: **

Let us assume that R as red ball and W as white ball.

Given in the question that white balls are identical so, if we pick any one of the three white balls is same.

So, the total number of spaces will be (2^2)-1 = 3, and they will be {WW, WR, RW}.

**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: **

There are 2 possible outcomes head(H) and tail(T) when you flip a coin. When you roll a die there are 6 possible outcomes 1, 2, 3, 4, 5, 6

When you get a head(H):

You should flip a coin another time when the sample space will be: {HT, HH}

When you get a tail(T):

You should die a roll where the sample space will be: {(T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}

The total sample space will be: {(HT), (HH), (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 the sample space of this experiment.**

**Solution: **

There are two possible outcomes either Defective(D) or Non-defective(N) and there are three bulbs now, the sample space will be 2^3 = 8.

And the possible outcomes will be: {DDD, DDN, DND, NDD, DNN, NDN, NND, NNN}

**Question 12. A coin is tossed. If the outcome 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: **

There are 2 possible outcomes head(H) and tail(T) when you flip a coin.

When you roll a die there are 6 possible outcomes 1,2,3,4,5,6

When the outcome is a head and die shows odd number the sample space will be:

{(H,1), (H,3), (H,5)}

When the outcome is a head and die shows even number the sample space will be:

Now, we need to roll a die again and the sample space will be of (1*3*6 = 18) and they are:

{(H,2,1), (H,2,2), (H,2,3), (H,2,4), (H,2,5), (H,2,6), (H,4,1), (H,4,2), (H,4,3), (H,2,4), (H, 4,5), (H,4,6), (H,6,1),

(H,6,2), (H,6,3), (H,6,4), (H,6,5), (H,6,6)}

When the outcome is tail: The outcome will be 1 i.e. (T).

The overall sample space will be 22 and they are:

{(H,1), (H,3), (H,5), (H,2,1), (H,2,2), (H,2,3), (H,2,4), (H,2,5), (H,2,6), (H,4,1), (H,4,2), (H,4,3), (H,2,4), (H,4,5),

(H,4,6), (H,6,1), (H,6,2), (H,6,3), (H,6,4), (H,6,5), (H,6,6), (T)}

**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:**

The numbers written on the slips are 1,2,3,4. Now if I am picking the first slip the probability will be 4 and I picked it and the box is left with 3 more slips and now, I am picking another slip where probability will be 3.

Therefore, the sample space will be

4*3 = 12 and they are:

{(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: **

There are 2 possible outcomes head(H) and tail(T) when you flip a coin.

When you roll a die there are 6 possible outcomes 1, 2, 3, 4, 5, 6

If the number on the die is even:

The sample space will be 6, and they are: {(2, H),(4,H),(6,H),(2,T),(4,T),(6,T)}

If the number on the die is odd:

We should roll the die twice and the sample space is 12 and they are:

{(1,H,H), (3,H,H), (5,H,H), (1,H,T), (3,H,T), (5,H,T), (1,T,H), (3,T,H), (5,T,H), (1,T,T), (3,T,T), (5,T,T)}

The total sample space will be:

{(2,H), (4,H), (6,H), (2,T), (4,T), (6,T), (1,H,H), (3,H,H), (5,H,H), (1,H,T), (3,H,T), (5,H,T), (1,T,H), (3,T,H), (5,T,H), (1,T,T), (3,T,T), (5,T,T)}

**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: **

There are 2 possible outcomes head(H) and tail(T) when you flip a coin.

Assume R1, R2 as two red balls and B1, B2, B3 as three black balls.

If the coin shows a tail:

The sample space will be 5, and they are: {(TR1), (TR2), (TB1), (TB2), (TB3)}

If the coin shows a head:

We throw a die. The sample space will be 5, and they are {(H,1), (H,2), (H,3), (H,4), (H,5), (H,6)}

The overall sample space will be:

{(T,R1), (T,R2), (T,B1), (T,B2), (T,B3), (H,1), (H,2), (H,3), (H,4), (H,5), (H,6)}

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

**Solution: **

When you roll a die there are 6 possible outcomes 1,2,3,4,5,6

When you get 6 on the first throw, the sample space will be: {6}

When you get 6 on the second throw, the sample space will be: {(1,6), (2,6), (3,6), (4,6), (5,6)}

When you get 6 on the third throw, the sample space will be: {(1,1,6),(1,2,6),(1,3,6),……}

The sample space is infinitely defined. This process can go infinite times.

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