Class 11 RD Sharma Solutions- Chapter 33 Probability - Exercise 33.1 | Set 2

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 a 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. 

Rooms	P	Q
Boys	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 blackballs. 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), ........... } 

Summary

Exercise 33.1 Set 2 covers probability concepts, including experimental probability, theoretical probability, and probability of events. Students learn to calculate probabilities using formulas and theorems. Understanding probability is crucial for statistics, data analysis, and real-world applications.