Class 11 RD Sharma Solutions- Chapter 33 Probability - Exercise 33.1 | Set 1

Question 1. A coin is tossed once. Write its sample space.

Solution: 

Since, a coin has two faces either head or tails. So, if the coin is tossed it will land on either of these two sides. Hence, sample space will be given as:

S = { H, T }, where H is the event Head comes upon landing and T is the event Tails comes upon landing.

Question 2. If a coin is tossed two times, describe the sample space associated with this experiment.

Solution: 

If a coin is tossed two times, then there can be four possible cases which are:

(A) both coins show head

(B) first coin shows head and second shows tail

(C) first coin shows tail and second shows head

(A) Both coins show tail

Hence, sample space will be given as:

S = { HH, HT, TH, TT }

Question 3. If a coin is tossed three times (or three coins are tossed together), then describe the sample space for this experiment.

Solution: 

If a coin is tossed three times (or three coins are tossed together), then there will be 2^3 = 8  possible cases which are:

(A) all coins show head

(B) first two coins show head and third shows tail

(C) first coin shows head and other two shows tail

(D) first coin shows tail and other two shows head

(E) first coin shows head second shows tail and third shows head

(F) first coin shows tail second shows head and third shows tail

(G) first  two coins show tail and third shows head

(H) all coins show tail

Hence, sample space will be given as:

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

Question 4. Write the sample space for the experiment of tossing a coin four times.

Solution:

If a coin is tossed four times, then there will be 2^4 = 16 possible cases and its sample space will be given as:

S = { HHHH, HHHT, HHTT, HHTH, HTHH, THHH, HTTH, HTHT, 

         TTTT, TTTH, TTHH, TTHT, THTT, HTTT, THHT, THTH }

Question 5. Two dice are thrown. Describe the sample space for this experiment.

Solution: 

Since, a dice has six faces, when a dice is thrown two times, there can be 6^2 = 36 possible cases, so the sample space can be written as:

S = { (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 6. What is the totalnumber of elementary events associated to the random experiment of throwing three dice together?

Solution: 

Since, a dice has six faces, when a dice is thrown three times, total  number of elementary events associated will be 6^3 = 216 events.

Question 7. A coin is tossed and then a die is thrown. Describe the sample space of this experiment.

Solution:

When a coin is tossed it can land on either of two sides Head or Tail, and when a dice is rolled, it can have six possible outcomes. So, there can be in total 2 x 6 = 12 possible events. Hence, the ample space can be written as:

S = { (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 8. A coin is tossed and then a die is rolled only in case a head is shown on the coin. Describe the sample space for this experiment.

Solution: 

When a coin is tossed it can land on either of two sides Head or Tail, if the head turns up then we will roll the dice as given in the question. So, there will be in total 1 + 1 x 6 = 7 possible events. Hence, the sample space can be given as:

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

Question 9. A coin is tossed twice. If the second throw results in a tail, a die is thrown. Describe the sample space for this experiment.

Solution: 

If a coin is tossed two times, then there can be four possible cases which are: HH, HT, TH, TT. Now, according to question, a dice is rolled only if second coin shows tails. So the total number of events associated with this event are:

2 x 6 + 2 = 14 events and sample space can be given as:

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

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

Question 10. An experiment consists of tossing a coin and then tossing it second time if head occurs. If a tail occurs on the first toss, then a die is tossed once. Find the sample space.

Solution: 

When a coin is tossed and head comes then the coin is tossed again, so the sample space will be:

S1 = { HH, HT, TH, TT }

and when the coin is tossed and tail comes then the die is tossed, so the sample space will be:

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

Therefore, sample space for the entire experiment can be written as a union of these two sample space as:

S = S1 ∪ S2 = { HH, HT, TH, TT,  (T,1), (T,2), (T,3), (T,4), (T,5), (T,6) }

Question 11. A coin is tossed. If it shows tail, we draw a ball from a box which contains 2 red 3 black balls; if it shows head, we throw a die. Find the sample space of this experiment.

Solution:

When a coin is tossed, we have two possible outcomes either head (H) or tail (T).

If head (H) turns up, we throw a die, then the sample space for this experiment is:

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

and if tail (T) turns up, then we draw a ball from the box containing 2 red and 3 black balls, sample space for this experiment is:

S2 = { (T,R1), (T,R2), (T,B1), (T,B2), (T,B3) }

Therefore, sample space for the entire experiment can be written as a union of these two sample space as:

S = S1 ∪ S2 = { (H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,R1), (T,R2), (T,B1), (T,B2), (T,B3) }

Question 12. A coin is tossed repeatedly until a tail comes up for the first time. Write the sample space for this experiment.

Solution:

According to the question, we will stop tossing the coin as soon as we get our first tail, else we will toss it repeatedly until we get a tail. Hence, the sample space for this experiment can be given as:

S = { T, HT, HHT, HHHT, ..... }

Summary

Exercise 33.1 Set 1 covers basic probability concepts, including experimental probability and theoretical probability. Students learn to calculate probabilities using formulas and theorems. Understanding probability is crucial for statistics, data analysis, and real-world applications.