\displaystyle \textbf{Question 1: }\text{A coin is tossed once. Write its sample space.}
\displaystyle \textbf{Answer:}
\displaystyle \text{When a coin is tossed once, there are two possible outcomes:}
\displaystyle \text{Head (H) or Tail (T).}
\displaystyle \therefore S=\{H,T\}.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{If a coin is tossed two times, describe the sample space}
\displaystyle \text{associated with this experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{A coin is tossed two times. Each toss has two possible outcomes:}
\displaystyle \text{Head (H) or Tail (T).}
\displaystyle \text{Hence, the total number of possible outcomes}=2^2=4.
\displaystyle \therefore S=\{HH,HT,TH,TT\}.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{If a coin is tossed three times (or three coins are tossed together),}
\displaystyle \text{describe the sample space for this experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{A coin is tossed three times.}
\displaystyle \text{Each toss has two possible outcomes: Head (H) or Tail (T).}
\displaystyle \text{Hence, the total number of possible outcomes}=2^3=8.
\displaystyle \therefore S=\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Write the sample space for the experiment of tossing a coin}
\displaystyle \text{four times.}
\displaystyle \textbf{Answer:}
\displaystyle \text{Each toss has two possible outcomes: Head (H) or Tail (T).}
\displaystyle \text{Hence, the total number of possible outcomes}=2^4=16.
\displaystyle \therefore S=\{HHHH,HHHT,HHTH,HHTT,HTHH,HTHT,HTTH,HTTT,
\displaystyle \phantom{\therefore S=\{}THHH,THHT,THTH,THTT,TTHH,TTHT,TTTH,TTTT\}.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Two dice are thrown. Describe the sample space of this}
\displaystyle \text{experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{When a die is rolled, there are six possible outcomes: }1,2,3,4,5\text{ and }6.
\displaystyle \text{Hence, when two dice are thrown, the total number of outcomes}=6^2=36.
\displaystyle \therefore S=\{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),
\displaystyle (2,1),(2,2),(2,3),(2,4),(2,5),(2,6),
\displaystyle (3,1),(3,2),(3,3),(3,4),(3,5),(3,6),
\displaystyle (4,1),(4,2),(4,3),(4,4),(4,5),(4,6),
\displaystyle (5,1),(5,2),(5,3),(5,4),(5,5),(5,6),
\displaystyle (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\}.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{What is the total number of elementary events associated}
\displaystyle \text{with the random experiment of throwing three dice together?}
\displaystyle \textbf{Answer:}
\displaystyle \text{Each die has }6\text{ possible outcomes.}
\displaystyle \text{Hence, when three dice are thrown together,}
\displaystyle \text{the total number of elementary events}=6^3=216.
\displaystyle \therefore \text{The total number of elementary events is }216.
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{A coin is tossed and then a die is thrown. Describe the}
\displaystyle \text{sample space for this experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{A coin has two possible outcomes: Head (H) or Tail (T).}
\displaystyle \text{A die has six possible outcomes: }1,2,3,4,5\text{ and }6.
\displaystyle \text{Hence, the total number of possible outcomes}=2\times6=12.
\displaystyle \therefore S=\{(H,1),(H,2),(H,3),(H,4),(H,5),(H,6),
\displaystyle \phantom{\therefore S=\{}(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)\}.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{A coin is tossed and then a die is rolled only in case a}
\displaystyle \text{head is shown on the coin. Describe the sample space}
\displaystyle \text{for this experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{If Head (H) occurs, the die is rolled and may show }1,2,3,4,5\text{ or }6.
\displaystyle \text{If Tail (T) occurs, the die is not rolled.}
\displaystyle \therefore S=\{(H,1),(H,2),(H,3),(H,4),(H,5),(H,6),T\}.
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{A coin is tossed twice. If the second throw results in a tail,}
\displaystyle \text{a die is thrown. Describe the sample space for this experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{When a coin is tossed twice, the possible outcomes are }
\displaystyle \{HH,HT,TH,TT\}.
\displaystyle \text{If the second toss is Tail, a die is rolled.}
\displaystyle \text{Hence, the outcomes }HT\text{ and }TT\text{ are followed by }1,2,3,4,5\text{ or }6.
\displaystyle \therefore S=\{HH,TH,(HT,1),(HT,2),(HT,3),(HT,4),(HT,5),(HT,6),
\displaystyle \phantom{\therefore S=\{}(TT,1),(TT,2),(TT,3),(TT,4),(TT,5),(TT,6)\}.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{An experiment consists of tossing a coin and then tossing it}
\displaystyle \text{a second time if Head occurs. If Tail occurs on the first toss,}
\displaystyle \text{then a die is tossed once. Find the sample space.}
\displaystyle \textbf{Answer:}
\displaystyle \text{If the first toss is Tail (T), a die is rolled.}
\displaystyle S_1=\{(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)\}.
\displaystyle \text{If the first toss is Head (H), the coin is tossed again.}
\displaystyle S_2=\{(H,H),(H,T)\}.
\displaystyle \therefore S=\{(T,1),(T,2),(T,3),(T,4),(T,5),(T,6),
\displaystyle \phantom{\therefore S=\{}(H,H),(H,T)\}.
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{A coin is tossed. If it shows Tail, a ball is drawn from a box}
\displaystyle \text{containing }2\text{ red and }3\text{ black balls. If it shows Head, a die is}
\displaystyle \text{thrown. Find the sample space of this experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{Let the two red balls be }R_1,R_2\text{ and the three black balls be }B_1,B_2,B_3.
\displaystyle \text{If Tail (T) occurs, one ball is drawn from the box.}
\displaystyle S_1=\{(T,R_1),(T,R_2),(T,B_1),(T,B_2),(T,B_3)\}.
\displaystyle \text{If Head (H) occurs, a die is thrown.}
\displaystyle S_2=\{(H,1),(H,2),(H,3),(H,4),(H,5),(H,6)\}.
\displaystyle \therefore S=\{(T,R_1),(T,R_2),(T,B_1),(T,B_2),(T,B_3),
\displaystyle \phantom{\therefore S=\{}(H,1),(H,2),(H,3),(H,4),(H,5),(H,6)\}.
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{A coin is tossed repeatedly until a Tail comes up for the}
\displaystyle \text{first time. Write the sample space for this experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{If Tail (T) occurs, the experiment ends.}
\displaystyle \text{If Head (H) occurs, the coin is tossed again.}
\displaystyle \text{This process continues until the first Tail appears.}
\displaystyle \therefore S=\{T,HT,HHT,HHHT,HHHHT,\ldots\}.
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{A box contains }1\text{ red and }3\text{ black balls. Two balls are}
\displaystyle \text{drawn at random in succession without replacement. Write the}
\displaystyle \text{sample space for this experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{The balls are distinct. Let the red ball be }R\text{ and the black balls be}
\displaystyle B_1,B_2\text{ and }B_3.
\displaystyle \text{Since the balls are drawn without replacement, order matters.}
\displaystyle \therefore S=\{(R,B_1),(R,B_2),(R,B_3),(B_1,R),(B_1,B_2),(B_1,B_3),
\displaystyle \phantom{\therefore S=\{}(B_2,R),(B_2,B_1),(B_2,B_3),(B_3,R),(B_3,B_1),(B_3,B_2)\}.
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{A pair of dice is rolled. If the outcome is a doublet, a coin is}
\displaystyle \text{tossed. Determine the total number of elementary events associated}
\displaystyle \text{with this experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{When a pair of dice is rolled, there are }6^2=36\text{ outcomes.}
\displaystyle \text{Among these, }6\text{ are doublets and }30\text{ are non-doublets.}
\displaystyle \text{Each non-doublet contributes one elementary event.}
\displaystyle \text{Each doublet is followed by a coin toss, giving }2\text{ elementary events.}
\displaystyle \therefore \text{Total number of elementary events}=30+(6\times2)=42.
\displaystyle \therefore \text{The required number of elementary events is }42.
\displaystyle \\

\displaystyle \textbf{Question 15: }\text{A coin is tossed twice. If the second toss results in a Head,}
\displaystyle \text{a die is rolled. Write the sample space for this experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{When a coin is tossed twice, the possible outcomes are }
\displaystyle \{HH,HT,TH,TT\}.
\displaystyle \text{If the second toss is Head, a die is rolled.}
\displaystyle \text{Hence, the outcomes }HH\text{ and }TH\text{ are followed by }1,2,3,4,5\text{ or }6.
\displaystyle \therefore S=\{HT,TT,(HH,1),(HH,2),(HH,3),(HH,4),(HH,5),(HH,6),
\displaystyle \phantom{\therefore S=\{}(TH,1),(TH,2),(TH,3),(TH,4),(TH,5),(TH,6)\}.
\displaystyle \\

\displaystyle \textbf{Question 16: }\text{A bag contains }4\text{ identical red balls and }3\text{ identical black}
\displaystyle \text{balls. One ball is drawn, replaced, and then another ball is drawn.}
\displaystyle \text{What are the possible outcomes of the experiment?}
\displaystyle \textbf{Answer:}
\displaystyle \text{Let }R\text{ denote a red ball and }B\text{ denote a black ball.}
\displaystyle \text{Since the first ball is replaced before the second draw, each draw has}
\displaystyle \text{the same two possible outcomes: }R\text{ or }B.
\displaystyle \therefore S=\{RR,RB,BR,BB\}.
\displaystyle \\

\displaystyle \textbf{Question 17: }\text{In a random sampling, three items are selected from a lot.}
\displaystyle \text{Each item is tested and classified as defective (D) or}
\displaystyle \text{non-defective (N). Write the sample space of this experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{Each of the three items may be defective (D) or non-defective (N).}
\displaystyle \text{Hence, the total number of possible outcomes}=2^3=8.
\displaystyle \therefore S=\{DDD,DDN,DND,DNN,NDD,NDN,NND,NNN\}.
\displaystyle \\

\displaystyle \textbf{Question 18: }\text{An experiment consists of the boy-girl composition of}
\displaystyle \text{families with }2\text{ children.}
\displaystyle \textbf{(i) }\text{What is the sample space if we are interested in knowing}
\displaystyle \text{whether it is a boy or girl in the order of their births?}
\displaystyle \textbf{(ii) }\text{What is the sample space if we are interested in the}
\displaystyle \text{number of boys in a family?}
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) Let }B\text{ denote a boy and }G\text{ denote a girl.}
\displaystyle \text{Since the order of birth is important, there are }2^2=4\text{ outcomes.}
\displaystyle \therefore S=\{BB,BG,GB,GG\}.
\displaystyle \text{Here, the first letter denotes the elder child and the second letter}
\displaystyle \text{denotes the younger child.}
\displaystyle \text{(ii) The possible numbers of boys in a family with two children are}
\displaystyle 0,1\text{ and }2.
\displaystyle \therefore S=\{0,1,2\}.
\displaystyle \\

\displaystyle \textbf{Question 19: }\text{There are three coloured dice: red, white and black. One die is}
\displaystyle \text{drawn at random and rolled. Its colour and the number on its}
\displaystyle \text{uppermost face are noted. Describe the sample space.}
\displaystyle \textbf{Answer:}
\displaystyle \text{Let }R,W\text{ and }B\text{ denote the red, white and black dice respectively.}
\displaystyle \text{Each die has six possible outcomes: }1,2,3,4,5\text{ and }6.
\displaystyle \therefore S=\{(R,1),(R,2),(R,3),(R,4),(R,5),(R,6),
\displaystyle \phantom{\therefore S=\{}(W,1),(W,2),(W,3),(W,4),(W,5),(W,6),
\displaystyle \phantom{\therefore S=\{}(B,1),(B,2),(B,3),(B,4),(B,5),(B,6)\}.
\displaystyle \\

\displaystyle \textbf{Question 20: }\text{Two boys and two girls are in room }P\text{ and one boy and}
\displaystyle \text{three girls are in room }Q.\text{ Write the sample space for the experiment}
\displaystyle \text{in which a room is selected and then a person.}
\displaystyle \textbf{Answer:}
\displaystyle \text{Let the persons in room }P\text{ be }B_1,B_2,G_1,G_2.
\displaystyle \text{Let the persons in room }Q\text{ be }B_3,G_3,G_4,G_5.
\displaystyle \therefore S=\{(P,B_1),(P,B_2),(P,G_1),(P,G_2),
\displaystyle \phantom{\therefore S=\{}(Q,B_3),(Q,G_3),(Q,G_4),(Q,G_5)\}.
\displaystyle \\

\displaystyle \textbf{Question 21: }\text{A bag contains one white and one red ball. A ball is drawn.}
\displaystyle \text{If the ball drawn is white, it is replaced and another ball is drawn.}
\displaystyle \text{Otherwise, a die is tossed. Write the sample space.}
\displaystyle \textbf{Answer:}
\displaystyle \text{Let }W\text{ denote the white ball and }R\text{ denote the red ball.}
\displaystyle \text{If the first ball is white, it is replaced and another ball is drawn.}
\displaystyle S_W=\{(W,W),(W,R)\}.
\displaystyle \text{If the first ball is red, a die is rolled.}
\displaystyle S_R=\{(R,1),(R,2),(R,3),(R,4),(R,5),(R,6)\}.
\displaystyle \therefore S=\{(W,W),(W,R),(R,1),(R,2),(R,3),(R,4),(R,5),(R,6)\}.
\displaystyle \\

\displaystyle \textbf{Question 22: }\text{A box contains }1\text{ white and }3\text{ identical black balls.}
\displaystyle \text{Two balls are drawn at random in succession without replacement.}
\displaystyle \text{Write the sample space for this experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{Let }W\text{ denote a white ball and }B\text{ denote a black ball.}
\displaystyle \text{Since there is only one white ball, }(W,W)\text{ is not possible.}
\displaystyle \therefore S=\{(W,B),(B,W),(B,B)\}.
\displaystyle \\

\displaystyle \textbf{Question 23: }\text{An experiment consists of rolling a die and then tossing a}
\displaystyle \text{coin once if the number on the die is even. If the number is odd,}
\displaystyle \text{the coin is tossed twice. Write the sample space.}
\displaystyle \textbf{Answer:}
\displaystyle \text{A die has three even outcomes }2,4,6\text{ and three odd outcomes }1,3,5.
\displaystyle \text{If an even number occurs, the coin is tossed once.}
\displaystyle \text{If an odd number occurs, the coin is tossed twice.}
\displaystyle \therefore S=\{(2,H),(2,T),(4,H),(4,T),(6,H),(6,T),
\displaystyle \phantom{\therefore S=\{}(1,HH),(1,HT),(1,TH),(1,TT),
\displaystyle \phantom{\therefore S=\{}(3,HH),(3,HT),(3,TH),(3,TT),
\displaystyle \phantom{\therefore S=\{}(5,HH),(5,HT),(5,TH),(5,TT)\}.
\displaystyle \\

\displaystyle \textbf{Question 24: }\text{A die is thrown repeatedly until a six comes up.}
\displaystyle \text{What is the sample space for this experiment?}
\displaystyle \textbf{Answer:}
\displaystyle \text{The experiment ends when }6\text{ appears for the first time.}
\displaystyle \text{Before the first }6,\text{ each throw can only be }1,2,3,4\text{ or }5.
\displaystyle \therefore S=\{6,(1,6),(2,6),(3,6),(4,6),(5,6),
\displaystyle \phantom{\therefore S=\{}(1,1,6),(1,2,6),(1,3,6),(1,4,6),(1,5,6),
\displaystyle \phantom{\therefore S=\{}(2,1,6),(2,2,6),(2,3,6),\ldots\}.
\displaystyle \\

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