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## Here's the Solution to this Question

$1)$

When two coins are tossed, the sample space $S$ is,

$S=\{HH,HT,TH,TT\}$

Where, $T$ is the event that a tail occurs and $H$ is the event that a head occurs.

Let the random variable $T$ denote the number of tails. From the sample space above, the random variable $T$ may take on the values 0,1,2. Therefore, $t=0,1,2$

$2)$

For each toss, the outcome belongs to one of the following: $\{H, T\}$

where $H$ is heads and $T$ is tails.

We want to count the total number of tails obtained from those 3 tosses.

Observe that any one of the following cases may happen:

1. $\{TTT\}$ - All the three outcomes are 'Tail'. Hence, T = 3 in this case.
2. $\{TTH\}$ - Any 2 of the 3 outcomes are 'Tail' and the remaining one is a 'Head' Hence, T = 2 in this case.
3. $\{THH\}$ - Any 1 of the 3 outcomes are 'Tail' and the remaining two are 'Head' Hence, T = 1 in this case.
4. $\{HHH\}$ - All the three outcomes are 'Head'. So the number of tails is 0. Hence, T = 0 in this case.

Note that, the 4 cases listed above explores all possible outcomes. Hence, the random variable T takes any one value from {0, 1, 2, 3}. That is, $t=0,1,2,3$

$3)$

Let the random variable $A$ be the event that we select an American and $G$ be the event that a German is selected.

The sample space when 3 consuls are randomly selected is,

$S=\{AAA,AAG,AGA,AGG,GAA,GAG,GGA,GGG\}$

From the sample points in the sample space above, the number of Germans selected vary from $0,1,2,3$.

However, the sample point $\{GGG\}$ consisting of three Germans is not possible because we only have 2 Germans.

Therefore, the random variable $G$ may take on the values 0,1,2. That is, $g=0,1,2$

$4)$

For each toss, the outcome belongs to one of the following:$\{H, T\}$

where $H$ is heads and $T$ is tails.

We want to count the total number of tails obtained from those 4 tosses.

Observe that any one of the following cases may happen:

1. $\{TTTT\}$ - All the four outcomes are 'Tail'. Hence, T = 4 in this case.
2. e.g. $\{TTTH\}$ - Any 3 of the 4 outcomes are 'Tail' and the remaining one is a 'Head' Hence, T = 3 in this case.
3. e.g. $\{TTHH\}$ - Any 2 of the 4 outcomes are 'Tail' and the remaining two are 'Head' Hence, T = 2 in this case.
4. e.g. $\{THHH\}$ - Any 1 of the 4 outcomes are 'Tail' and the remaining three are 'Head' Hence, T = 1 in this case.
5. $\{HHHH\}$ - All the four outcomes are 'Head'. So the number of tails is 0. Hence, T = 0 in this case.

Note that, the 5 cases listed above explores all possible outcomes. Hence, the random variable $T$  takes any one value from {0, 1, 2, 3, 4}. That is $t=\{0,1,2,3,4\}$

$5)$

For each rolled dice, the outcome belongs to one of the following: {1, 2, 3, 4, 5, 6}

Thus we can define the following sample space for a two balanced dice:

(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)

Therefore the sum of the number of dots that will appear S takes any one value from {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

$6)$

Let X be the number of boys in a family of four children:

Observe that any one of the following cases may happen:

1. $\{BBBB\}$ - All the four outcomes are 'Boy'. Hence, $X = 4$  in this case.
2. e.g. $\{BBBG\}$ - Any 3 of the 4 outcomes are 'Boy' and the remaining one is a 'Girl' Hence, $X = 3$ in this case.
3. e.g. $\{BBGG\}$ - Any 2 of the 4 outcomes are 'Boy' and the remaining two are 'Girl' Hence, $X = 2$ in this case.
4. e.g. $\{BGGG\}$ - Any 1 of the 4 outcomes are 'Boy' and the remaining three are 'Girl' Hence, $X = 1$  in this case.
5. $\{GGGG\}$ - All the four outcomes are 'Girl'. So the number of boys is 0. Hence, $X = 0$ in this case.

The 5 cases listed above explores all possible outcomes. Hence, the random variable X takes any one value from {0, 1, 2, 3, 4}.

$7)$

Let $G$ and $B$ be the events that green and blue dice are chosen at random. The sample space for choosing three dice is,

$S=\{GGG,GGB,GBG,GBB,BGG,BGB,BBG,BBB\}$

From the sample space above, sample points with,

$i)$

$G=1$ are,

$GBB \\ BGB\\ BBG$

$ii)$

$G=2$ are,

$BGG\\ GBG\\ GGB$

$iii)$

$G=3$ is,

$GGG$

$iv)$

The sample point, $BBB$ is where, $G=0$.

Therefore, the random variable $G$ may take on the values, 0,1,2,3.