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

When a coin is tossed, there are two possibilities. It could either turn up Head (H) or Tail (T).

Hence the sample space will be $\displaystyle S = \{ H, T \}$ $\\$

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

Given: If Coin is tossed twice times. In each toss there can be two outcomes. It could either turn up Head (H) or Tail (T).

We know that, the coins are tossed two times, and then the total number of possible outcomes is $\displaystyle 2^2 = 4$

Hence the sample space will be $\displaystyle S = \{ HT, TH, HH, TT \}$ $\\$

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

Given: If a coin is tossed three times.

We know that, the coins are tossed three times, and then the total number of possible outcomes is $\displaystyle 2^3 = 8$

Hence the sample space will be $\displaystyle S = \{ HHH, TTT, HHT, HTH, THH, HTT, THT, TTH \}$ $\\$

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

When a coin is tossed, there are two possibilities. It could either turn up Head (H) or Tail (T).

We know that, the coins are tossed four times, and then the total number of possible outcomes is $\displaystyle 2^4 = 16$

Hence the sample space will be $\displaystyle S = \{ HHHH, TTTT, HHHT, HHTH, HTHH, THHH, HHTT, \\ HTTH, HTHT, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH \}$ $\\$

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

When a dice is rolled, there are six possibilities. It could either turn up $\displaystyle 1, 2, 3, 4,5 \text{ or } 6.$

We know that, when two dice are rolled , and then the total number of possible outcomes is $\displaystyle 6^2 = 36$

Hence the sample space will be $\displaystyle S = \{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), \\ (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), \\ (5, 1), (5, 2), (5, 3), (5, 4), (5, 6), (5, 5), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \}$ $\\$

Question 6: What is the total number of elementary events associated to the random experiment of
throwing three dice together?

Given: Three dice is rolled together.

We know that, three dice are thrown together.

When a dice is rolled, there are six possibilities. It could either turn up $\displaystyle 1, 2, 3, 4,5 \text{ or } 6.$

So, the total numbers of elementary event on throwing three dice are $\displaystyle 6^3 = 216$

Therefore the total number of elementary events are $\displaystyle 216$ $\\$

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

Given: A coin is tossed and then a die is thrown.

We know that, the coin is tossed and die is thrown.

When a coin is tossed, there are two possibilities. It could either turn up Head (H) or Tail (T).

When a dice is rolled, there are six possibilities. It could either turn up $\displaystyle 1, 2, 3, 4,5 \text{ or } 6.$

Therefore, The total number of Sample space together is $\displaystyle 2 \times 6 = 12$

Hence the sample space will be $\displaystyle S = \{ (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), \\ \\ (T, 5), (T, 6) \}$ $\\$

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.

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

When a coin is tossed, there are two possibilities. It could either turn up Head (H) or Tail (T).

When a dice is rolled, there are six possibilities. It could either turn up $\displaystyle 1, 2, 3, 4,5 \text{ or } 6.$

Hence the sample space will be $\displaystyle S = \{ (H,1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), T \}$ $\\$

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.

Given: A coin is tossed twice. If the second throw results in a tail, a die is thrown.

When a coin tossed twice, then sample spaces for only coin will be: $\displaystyle \{ HH, TT, HT, TH \}$

Now if the second throw results in a tail, a die is thrown. Therefore when We get $\displaystyle TT \text{ or } HT,$ dice is rolled.

When a dice is rolled, there are six possibilities. It could either turn up $\displaystyle 1, 2, 3, 4,5 \text{ or } 6.$

Hence the sample space will be $\displaystyle S = \{ HH, TH, (HT, 1), (HT, 2), (HT, 3), (HT, 4), (HT, 5), (HT, 6), \\ (TT, 1), (TT, 2), (TT, 3), (TT, 4), (TT, 5), (TT, 6) \}$ $\\$

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.

In the given experiment, coin is tossed and if the outcome is tail then, die will be rolled.

When a coin is tossed, there are two possibilities. It could either turn up Head (H) or Tail (T).

When a dice is rolled, there are six possibilities. It could either turn up $\displaystyle 1, 2, 3, 4,5 \text{ or } 6.$

If the outcome for the coin is tail then sample space is $\displaystyle S1= \{ (T, 1)(T, 2)(T, 3)(T, 4)(T, 5)(T, 6) \}$

If the outcome is head then the sample space is $\displaystyle S2 =\{ (H, H)(H, T) \}$

Hence the sample space will be $\displaystyle S = \{ (T, 1)(T, 2)(T, 3)(T, 4)(T, 5)(T, 6)(H, H)(H, T) \}$ $\\$

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.

When a coin is tossed, there are two possibilities. It could either turn up Head (H) or Tail (T).

Let the two red balls be R1 and R2 and the three black balls be B1, B2 and B3

According to question, If tail turned up, the a ball is drawn from a box.

So, Sample for This experiment $\displaystyle S1 = \{ (T, R1)(T, R2)(T, B1)(T, B2)(T, B3) \}$

Now, If Head is turned up, then die is rolled.

When a dice is rolled, there are six possibilities. It could either turn up $\displaystyle 1, 2, 3, 4,5 \text{ or } 6.$

So, Sample space for this experiment $\displaystyle S2= \{ (H, 1)(H, 2)(H, 3)(H, 4)(H, 5)(H, 6) \}$

Hence the sample space will be $\displaystyle S= \{ (T, R1), (T, R2), (T, B1), (T, B2), (T, B3), (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6) \}$ $\\$

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

Given: A coin is tossed repeatedly until Tail comes up for the first time.

In the given Experiment, a coin is tossed and if the outcome is tail the experiment is over.

And, if the outcome is Head then the coin is tossed again.

In the second toss also if the outcome is tail then experiment is over, otherwise coin is tossed again.

This process continues indefinitely

Hence the sample space will be $\displaystyle S = \{ T, HT, HHT, HHHT, HHHHT, ... \}$ $\\$

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.

Note: You cannot assume that the balls are identical.

Let us denote the red ball as R and the three black balls as B1, B2, and B3.

Hence the sample space will be $\displaystyle S = \{ (R, B1), (R, B2), (R, B3), (B1, R), (B1, B2), (B1, B3), (B2, R), \\ (B2, B1), (B2, B3), (B3, R), (B3, B1), (B3, B2) \}$ $\\$

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 to this experiment.

Given a pair of dice is rolled. If the outcome is a doublet, a coin is tossed.

A pair of dice is rolled,  Then, No. of elementary events are $\displaystyle 6^2 = 36$

Now, If outcomes is doublet means $\displaystyle (1, 1)(2, 2)(3, 3)(4, 4)(5, 5)(6, 6),$ then a coin is tossed.

If coin is tossed then no. of sample spaces is $\displaystyle 2.$

So, The total no. of elementary events including doublet $\displaystyle = 6 \times 2=12$

Thus, The Total number of elementary events are $\displaystyle 30+12 = 42$

Hence, $\displaystyle 42$ events will occur for this experiments. $\\$

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.

A coin is tossed twice , So the outcomes are $\displaystyle S1 = {HH, HT, TH, TT}$

Now, If the second drawn result is head , the a die is rolled.

Then the elementary events is $\displaystyle S2 = \{ (HH, 1), (HH, 2)(HH, 3), (HH, 4)(HH, 5), (HH, 6), \\ (HH, 1), (TH, 2)(TH, 3), (TH, 4)(TH, 5), (TH, 6) \}$

Hence the sample space will be $\displaystyle S = \{ (HH), (HT), (TH), (TT)(HH, 1), (HH, 2)(HH, 3), (HH, 4) \\ (HH, 5), (HH, 6), (HH, 1), (TH, 2)(TH, 3), (TH, 4)(TH, 5), (TH, 6) \}$ $\\$

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?

Given a bag contains 4 identical red balls and 3 identical black balls.

Let us Assume Red $\displaystyle = R$

Let us Assume Black $\displaystyle = B$

Now, A ball is drawn in first attempt, So elementary events is $\displaystyle S1=\{ R, B \}$

And, The ball will put into the bag and draw again, then elementary events are $\displaystyle S2= \{ R, B \}$

Thus, The total sample space associated is $\displaystyle S=\{ RR, RB, BR, BB \}$ $\\$

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 of this experiment.

In the random sampling , three items are selected so it could be

(a) All defective (D)

(b) All non-defective(N)

(c) Combination of both defective and non defective  We have 2 category (N and D) and we have three condition

So, The number of sample is $\displaystyle 2^3 = 8$

Sample space associated with this experiment is $\displaystyle S= \{ DDD, NDN, DND, DNN, NDD, DDN, NND, NNN \}$ $\\$

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?

(i) Here, The family has 2 children

Let us Assume Boy = B  Let us Assume Girl = G

So, The number of sample spaces for 2 children is $\displaystyle 2^2=4$

Sample space are, $\displaystyle S=\{ (B1, B2), (G1, G2), (G1, B2), (B1, G2) \}$

Where , number 1 and 2 are represent elder and younger.

(ii) Here, The family has 2 children, So the possibility of boys in a family is:

(a) No boys only girl

(b) One boy and one girl

(c) Two boys only

So, The Sample space for the given condition is: $\displaystyle S=\{ 0, 1, 2 \}$ $\\$

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

When a dice is rolled, there are six possibilities. It could either turn up $\displaystyle 1, 2, 3, 4,5 \text{ or } 6.$

Let us Assume Red = R      White = W     Black = B

According to question,  Dice is selected , then rolled its color and number is noted.

Firstly, selected Red Dice then, Possible samples are: $\displaystyle S_R=\{ (R, 1), (R, 2)(R, 3), (R, 4), (R, 5), (R, 6) \}$

Then, White Dice will be selected , Possible samples are: $\displaystyle S_W=\{ (W, 1), (W, 2), (W, 3), (W, 4), (W, 5), (W, 6) \}$

Lastly, Black Dice will be selected and rolled, Possible samples are: $\displaystyle S_B= \{ (B, 1)(B, 2)(B, 3)(B, 4)(B, 5), (B, 6) \}$

Thus, The total sample for the given experiment is $\displaystyle S= \{ (B, 1)(B, 2)(B, 3)(B, 4)(B, 5), (B, 6), \\ (W, 1), (W, 2), (W, 3), (W, 4), (W, 5), (W, 6), \\ (R, 1), (R, 2)(R, 3), (R, 4), (R, 5), (R, 6) \}$ $\\$

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

Let us denote two boys and girls in room $\displaystyle P$ as $\displaystyle B_1 , B_2$ and $\displaystyle G_1 , G_2$ respectively.

Let us denote 1 boy and 3 girls in room $\displaystyle Q$ as $\displaystyle B_3$ and $\displaystyle G_3, G_4, G_5$ respectively.

Sample spaces for room P are $\displaystyle S_p= \{ (P, B_1), (P, B_2), (P, G_1), (P, G_2) \}$

Sample spaces for room Q are $\displaystyle S_q= \{ (Q, B_3), (Q, G_1), (Q, G_2), (Q, G_3) \}$

Now, The total sample spaces for the given experiment: $\displaystyle S = \{ (P, B_1), (P, B_2), (P, G_1), (P, G_2), (Q, B_3), (Q, G_1), (Q, G_2), (Q, G_3) \}$ $\\$

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 dice is tossed. Write the sample space for this experiment.

There is 1 white ball , 1 red ball in a bag.

Let us denote White ball with W and Red Ball with R.

When one ball is drawn, it may be a White or Red.

So, The sample space of drawing on white ball with replacement $\displaystyle S_w=\{ (W, W), (W, R) \}$

Again, If red ball is drawn , a dice is rolled

So, The sample space for red ball with dice $\displaystyle S_r=\{ (R, 1), (R, 2), (R, 3), (R, 4), (R, 5), (R, 6) \}$

Thus, The required sample space for the experiment is: $\displaystyle S= {(W, W), (W, R), (R, 1), (R, 2), (R, 3), (R, 4), (R, 5), (R, 6)}$ $\\$

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.

Let R stands for red ball and W stands for white ball.

Then required sample space $\displaystyle S = \{ (R,W), (W,R), (W,W) \}$ $\\$

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.

A dice has 6 faces that are numbered from 1 to 6.

And, we know, 2, 4, 6 are even numbers and 1, 3, 5 are odd numbers.

Now, A coin has two face, Head(H) and Tail(T)

Therefore the total sample spaces by combining both experiments is $\displaystyle S= \{ (2, H), (2, T), (4, H), (4, T), (6, H), (6, T), (1, HH), (1, HT), (1, TH), \\ (1, TT), (3, HH), (3, HT) , (3, TH), (3, TT), (5, HH), (5, HT), (5, TH), (5, TT) \}$ $\\$

Question 24: A die is thrown repeatedly until a six comes up. What is the sample space for this experiment $\displaystyle S= \{ 6, (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (1, 1, 6), (1, 2, 6), (1, 3, 6), (1, 4, 6), \\ (1, 5, 6), (2, 1, 6), (2, 2, 6), (2, 3, 6), (2, 4, 6), (2, 5, 6) ... ... \}$