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: $S = \{H1, H2, H3, H4, H5, H6, T\}$