1   Describe the sample space for the indicated experiment: $A$ coin is tossed three times.

Solution :

$A$ coin has two faces: head $(H)$ and tail $(T).$$\\$ When a coin is tossed three times, the total number of possible outcome is $2^ 3 = 8$$\\$ Thus, when a coin is tossed three times, the sample space is given by: $S = \{\text{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}\}$

2   Describe the sample space for the indicated experiment: $A$ die is thrown two times.

Solution :

When a die is thrown, the possible outcomes are $1, 2, 3, 4, 5,$ or $6.$$\\$ When a die is thrown two times, the sample is given by $S = \{ ( x , y ) : x, y = 1, 2, 3, 4, 5, 6\}$ The number of elements in this sample space is $6 × 6 = 36,$ while the sample space is given by: $S = \{ (1,1 ) , ( 1, 2 ) , ( 1,3 ) , ( 1, 4 ) , ( 1,6 ) , ( 2,1 ) , ( 2, 2 ) , ( 2,3 ) , (2, 4 ) , ( 2,5 ) , ( 2,6 ) , ( 3,1 ) ,\\ ( 3, 2 ) , ( 3,3 ) , ( 3,4 ) , ( 3,5 ) , ( 3,6 ) , ( 4,1 ) , ( 4, 2 ) , (4,3 ) , ( 4, 4 ) , ( 4,5 ) , ( 4,6 ) , ( 5,1 ) , ( 5, 2 ) ,\\ ( 5,3 ) , ( 5, 4 ) , ( 5,5 ) , ( 5,6 ) , ( 6,1 ) , ( 6, 2 ) , ( 6,3 ) , ( 6, 4 ) , ( 6,5 ) , ( 6,6 )\}$

3   Describe the sample space for the indicated experiment: $A$ coin is tossed four times.

Solution :

When a coin is tossed once, there are two possible outcomes: head $(H)$ and tail $(T).$ When a coin is tossed four times, the total number of possible outcomes is $2^ 4 = 16$$\\$ Thus, when a coin is tossed four times, the sample space is given by: $S= \{\text{HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH,THTT, TTHH, TTHT, TTTH, TTTT}\}$

4   Describe the sample for the indicated experiment: $A$ coin is tossed and a die is thrown.

Solution :

A coin has two faces: head $(H)$ and tail $(T).$$\\$ A die has six faces that are numbered from $1$ to $6,$ with one number on each face.$\\$ Thus, when a coin is tossed and a die is thrown, the sample is given by: $S = \{ H 1, H 2, H 3, H 4, H 5, H 6, T 1, T 2, T 3, T 4, T 5, T 6 \}$

5   Describe the sample space for the indicated experiment: $A$ coin is tossed and then a die is rolled only in case a head is shown on the coin.

Solution :

A coin has two faces: head $(H)$ and tail $(T).$$\\$ A die has six faces that are numbered from $1$ to $6,$ with one number on each face.$\\$ Thus, when a coin is tossed and then a die is rolled only in case a head is shown on the coin, the sample space is given by: $S = \{H1, H2, H3, H4, H5, H6, T\}$

6   $2$ boys and $2$ girls are in Room X, and $1$ boy and $3$ girls in Room Y. Specify the sample space for the experiment in which a room is selected and then a person.

Solution :

Let us denote $2$ boys and $2$ girls in room $X$ as $B_ 1 , B_ 2$ and $G _1$ , and $G _2$ respectively. Let us denote $1$ boy and $3$ girls in room Y as $B _3 $, and $G_ 3 , G_ 4 , G_ 5$ respectively.$\\$ Accordingly, the required sample space is given by $S = \{ XB_ 1 , XB_ 2 , XG_ 1 , XG_ 2 , YB_ 3 , YG_ 3 , YG_ 4 , YG_ 5 \}$

7   One die of red colour, one of white colour and one of blue colour are placed in a bag. One die is selected at random and rolled, its colour and the number on its uppermost face is noted. Describe the sample space.

Solution :

A die has six faces that are numbered from $1$ to $6$, with one number on each face.$\\$ Let us denote the red, white, and blue dices as $R, W,$ and $B $respectively.$\\$ According, when a die is selected and then rolled, the sample is given by $S = \{R_1, R_2, R_3, R_4, R_5, R_6, W_1, W_2, W_3, W_4, W_5, W_6, B_1, B_2, B_3, B_4, B_5, B_6\}$

8   An experiment consists of recording 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 girls in the family?

Solution :

(i) When the order of the birth of a girl or a boy is considered, the sample space is given by $S = \{GG, GB, BG, BB\}$$\\$ (ii) Since the maximum number of children in each family is $2$, a family can either have $2$ girls or $1$ girl or no girl.$\\$ Hence, the required sample space is $S = \{0, 1, 2\}$$\\$

9   A box contains $1$ red and $3$ identical white balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.

Solution :

It is given that the box contains $1$ red ball and $3$ identical white balls. Let us denote the red ball with R and a while ball with W.$\\$ When two balls are drawn at random in succession without replacement, the sample space is given by $S = \{RW, WR, WW\}$

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

Solution :

A coin has two faces: head (H) and tail (T).$\\$ A die has six faces that are numbered from $1$ to $6$, with one number on each face.$\\$ Thus, in the given experiment, the sample space is given by $S = \{HH, HT, T1, T2, T3, T4, T5, T6\}$