**Question 13. A box contains 1 red and 3 black balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.**

**Solution:**

The box contains 1 red and 3 black balls and two balls are drawn without replacement , so the sample space associated with this event can be given as:

S = { (R,B1), (R,B2), (R,B3), (B1,R), (B1,B2), (B1,B2), (B2,R), (B2,B1), (B2,B3), (B3,R), (B3,B1), (B3,B2) }

**Question 14. A pair of dice is rolled. If the outcome is doublet, a coin is tossed. Determine the total number of elementary events associated with the experiment.**

**Solution: **

When a pair of dice is rolled, then there are in total 6 x 6 = 36 possible outcomes.

The term doublet refers to the event when the pair of dice after rolling has outcomes as (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), when a double is obtained then again the coin is tossed and we have outcome as either head (H) or tail (T).

Therefore, total number of elementary events = (36-6) + 6 x 2 = 30 + 12 = 42.

**Question 15. A coin is tossed twice. If the second draw results in a head, a die is rolled. Write the sample space for this experiment.**

**Solution: **

When two coins are tossed, then we have four possible outcomes as HH, HT, TH, TT. Now for those cases where in second draw head comes, we throw a die, then the sample space is written as:

S’ = { (HH,1), (HH,2), (HH,3), (HH,4), (HH,5), (HH,6),

(TH,1), (TH,2), (TH,3), (TH,4), (TH,5), (TH,6) }

Therefore, sample space for the entire experiment can be written as:

S = { (HT), (TT). (HH,1), (HH,2), (HH,3), (HH,4), (HH,5), (HH,6), (TH,1), (TH,2), (TH,3), (TH,4), (TH,5), (TH,6) }

**Question 16. A bag contains 4 identical red balls and 3 identical black balls. The experiment consists of drawing one ball, then putting it into the bag and again drawing a ball. What are the possible outcomes of the experiment.**

**Solution: **

Since, we have identical balls inside the bag, we can denote each red ball using a common notation as R and similarly each black ball can be denoted using symbol B.

So, after first draw the sample space will be S1 = {R,B}, the ball is again put back in the bag, so again for second draw sample space will be S2 = {R,B}.

Hence, sample space for the entire event is S = { RR, RB, BR, BB }

**Question 17. In a random sampling three items are selected from a lot. Each item is tested and classified as Defective (D)or Non-defective (N). Write the sample space for this experiment.**

**Solution: **

Three items stored in the lot can be: (a) all defective (b) all non-defective (c) a mixture of both defective and non-defective items.

Therefore, the possible sample space associated with this experiment can be given as:

S = {DDD, DDN, DND, NDD, NNN, NND. NDN, DNN }

**Question 18. An experiment consists of 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 boys in a family?**

**Solution: **

According to the question, if a family consists of two children then sample space can be given as:

(i) S = { (B1,B2), (B1,G2), (G1,B2), (G1,G2) }, the number represents the first and second child.

(ii) Since, there can be at most two children, there are three possibilities:

a) the family has 0 boys

b) the family has 1 boy

c) the family has 2 boys

Hence, the sample space S = {0,1,2}

**Question 19. There are three colored dice of red, white and black color. These dice are places in a bag. One dice is drawn at random from the bag and rolled, its color and the number on its face is noted describe the sample space for the experiment.**

**Solution: **

If we pick red colored dice and draw its sample space can be given as:

S1 = { (R,1), (R,2), (R,3), (R,4), (R,5), (R,6) }

similarly, If we pick red colored dice and draw its sample space can be given as:

S2 = { (B,1), (B,2), (B,3), (B,4), (B,5), (B,6) }

similarly, If we pick white colored dice and draw its sample space can be given as:

S3 = { (W,1), (W,2), (W,3), (W.4), (W,5), (W,6) }

Hence, sample space for the entire experiment = S1 U S2 U S3

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

(B,1), (B,2), (B,3), (B,4), (B,5), (B,6),

(W,1), (W,2), (W,3), (W.4), (W,5), (W,6) }

**Question 20. 2 boys and 2 girls are in a room P and 1 boy 3 girls in room Q. Write the sample space for the experiment in which a room is selected and then a person.**

**Solution: **

There are in total 2 rooms.

RoomsPQBoys 2 1 Girls 2 3 We can select a room in two ways: either P or Q, also selecting a person from a room can be done in from P in 4 ways. Similarly, from Q it can be done in 4 ways.

Therefore, sample space for this experiment can be written as:

S = { (P,B1), (P,B2), (P,G1), (P,G2),

(Q,B3), (Q,G3), (Q,G4), (Q,G5) }

**Question 21. A bag contains one white and one red ball. A ball is drawn from the bag. If the ball drawn is white it is replaced in the bag and again a ball is drawn. Otherwise, a die is tossed. Write the sample space for this experiment.**

**Solution: **

Out of two balls, if we draw a ball, it will be either red (R) or white (W).

When a white ball is drawn, it is replaced and then again a ball is drawn, therefore sample space

S1 = { (W,W), (W,R) }

Also, if a red ball is drawn then a die is rolled, therefore sample space

S2 = { (R,1), (R,2), (R,3), (R,4), (R,5), (R,6) }

Hence, sample space for the entire experiment, S = S1 U S2

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

**Question 22. A box contains 1 white and 3 identical black** **balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.**

**Solution: **

Since, we have identical black balls inside the box, we can denote each black ball using a common notation as B. Now, sample space for drawing two balls without replacement can be written as:

S = { (W,B), (B,W), (B,B) }

**Question 23. 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: **

Sample space for throwing a die:

S1 = { 1, 2, 3, 4, 5, 6 }

If the even number turns up on the dice, then a coin is tossed, so

S2 = { (2,H), (2,T), (4,H), (4,T), (6,H), (6,T) }

whereas when an odd number turns up on the dice, then a coin is tossed two times, so

S3 = { (1,HH), (1,HT), (1,TH), (1,TT), (3,HH), (3,HT),(3,TH), (3,TH), (5,HH), (5,HT), (5,TH), (5,TT) }

Therefore, sample space for the entire experiment, S = S2 U S3

S = { (2,H), (2,T), (4,H), (4,T), (6,H), (6,T),

(1,HH), (1,HT), (1,TH), (1,TT), (3,HH), (3,HT),

(3,TH), (3,TH), (5,HH), (5,HT), (5,TH), (5,TT) }

**Question 24. A die is thrown repeatedly until a six comes up. What is the sample space for this experiment.**

**Solution: **

According to the question the die keeps on rolling till we not get a six. So, the sample space can be written as:

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