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## Here's the Solution to this Question

a.There are $2^3=8$ possible outcomes

$S=\{DDD, DDN, DND, NDD,$

$DNN,NDN, NND, NNN\}$

b. The possible values of the random variable $X$ are $0, 1, 2, 3.$

We will assume that the probability of getting heads and tails is the same:

$p = q =1/2$$\def\arraystretch{1.5} \begin{array}{c:c} Possible \ Outcomes & A \\ \hline DDD & 3 \\ \hdashline DDN & 2 \\ \hdashline DND & 2 \\ \hdashline NDD & 2 \\ \hdashline DNN& 1 \\ \hdashline NDN & 1 \\ \hdashline NND & 1 \\ \hdashline NNN & 0 \\ \hdashline \end{array}$

c. Construct the probability distribution of the random variable

$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 & 3 \\ \hline p(x) & 1/8 & 3/8 & 3/8 & 1/8 \end{array}$

or

$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 & 3 \\ \hline p(x) & 0.125 & 0.375 & 0.375 & 0.125 \end{array}$

d.

$P(X=0)+P(X=1)+P(X=2)+P(X=3)$

$=1/8+3/8+3/8+1/8=1$

e.

The probability of each value of a discrete random variable is between 0 and 1, and the sum of all the probabilities is equal to 1.

a.There are $2^3=8$ possible outcomes

$S=\{DDD, DDN, DND, NDD,$

$DNN,NDN, NND, NNN\}$

b. The possible values of the random variable $X$ are $0, 1, 2, 3.$

We will assume that the probability of getting heads and tails is the same:

$p = q =1/2$$\def\arraystretch{1.5} \begin{array}{c:c} Possible \ Outcomes & A \\ \hline DDD & 3 \\ \hdashline DDN & 2 \\ \hdashline DND & 2 \\ \hdashline NDD & 2 \\ \hdashline DNN& 1 \\ \hdashline NDN & 1 \\ \hdashline NND & 1 \\ \hdashline NNN & 0 \\ \hdashline \end{array}$

c. Construct the probability distribution of the random variable

$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 & 3 \\ \hline p(x) & 1/8 & 3/8 & 3/8 & 1/8 \end{array}$

or

$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 & 3 \\ \hline p(x) & 0.125 & 0.375 & 0.375 & 0.125 \end{array}$

d.

$P(X=0)+P(X=1)+P(X=2)+P(X=3)$

$=1/8+3/8+3/8+1/8=1$

e.

The probability of each value of a discrete random variable is between 0 and 1, and the sum of all the probabilities is equal to 1.

a.There are $2^3=8$ possible outcomes

$S=\{DDD, DDN, DND, NDD,$

$DNN,NDN, NND, NNN\}$

b. The possible values of the random variable $X$ are $0, 1, 2, 3.$

We will assume that the probability of getting heads and tails is the same:

$p = q =1/2$$\def\arraystretch{1.5} \begin{array}{c:c} Possible \ Outcomes & A \\ \hline DDD & 3 \\ \hdashline DDN & 2 \\ \hdashline DND & 2 \\ \hdashline NDD & 2 \\ \hdashline DNN& 1 \\ \hdashline NDN & 1 \\ \hdashline NND & 1 \\ \hdashline NNN & 0 \\ \hdashline \end{array}$

c. Construct the probability distribution of the random variable

$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 & 3 \\ \hline p(x) & 1/8 & 3/8 & 3/8 & 1/8 \end{array}$

or

$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 & 3 \\ \hline p(x) & 0.125 & 0.375 & 0.375 & 0.125 \end{array}$

d.

$P(X=0)+P(X=1)+P(X=2)+P(X=3)$

$=1/8+3/8+3/8+1/8=1$

e.

The probability of each value of a discrete random variable is between 0 and 1, and the sum of all the probabilities is equal to 1.