91Ó°ÊÓ

Probability refers to the likelihood of a random event's outcome. This word refers to determining the likelihood of a given occurrence occurring.

The complement rule is a statistical theorem that establishes a link between the probability of an occurrence and the probability of its complement, such that if one of these probabilities is known, automatically the other is also known.

It is given that\({\rm{P(}}\)successful\({\rm{) = p}}\).

A packet can be transmitted at most\({\rm{10}}\)times –\({\rm{k}} \le {\rm{10}}\).

The transmissions are independent.

Probability Mass Function –

\({\rm{X = }}\)the number of times a packet is transmitted, thus\({\rm{X}}\)is the number of trials until the first success\({\rm{Y}}\)(successful transmission) occurs up to\({\rm{10}}\)packets.

The number of trials until the first success follows a geometric distribution with probability of success\(p\).

\({\rm{Y}} \sim {\rm{Geometric(p)}}\)

Definition geometric probability –

\({\rm{P(Y = k) = }}{{\rm{q}}^{{\rm{k - 1}}}}{\rm{p = (1 - p}}{{\rm{)}}^{{\rm{k - 1}}}}{\rm{p}}\)

The definition of geometric probability is for all nonnegative integers, however for \({\rm{X}}\) it is given \({\rm{k}} \le {\rm{10}}\). Use the definition for \({\rm{k = 1, 2, 3, 4, 5, 6, 7, 8, 9}}\).

The complement rule is represented as –

\({\rm{P(not A) = 1 - P(A)}}\)

Addition rule for disjoint or mutually exclusive events –

\({\rm{P(A or B) = P(A) + P(B)}}\)

Use the complement rule and the addition rule –

\(\begin{array}{c}{\rm{P(X = 10) = P(Y}} \ge {\rm{10) = 1 - P(Y}} \le {\rm{9) = 1 - P(Y = 1) - P(Y = 2) - \ldots - P(Y = 9)}}\\{\rm{ = 1 - }}\sum\limits_{{\rm{k = 0}}}^{\rm{9}} {\rm{P}} {\rm{(Y = k) = 1 - }}\sum\limits_{{\rm{k = 0}}}^{\rm{9}} {{{{\rm{(1 - p)}}}^{{\rm{k - 1}}}}} {\rm{p}}\end{array}\)

Combining these results to obtain the probability distribution –

\(\begin{array}{l}{\rm{p(k) = P(X = k)}}\\{\rm{ = }}\left\{ {\begin{array}{*{20}{c}}{{{{\rm{(1 - p)}}}^{{\rm{k - 1}}}}{\rm{p}}}&{{\rm{k = 1,2,3,4,5,6,7,8,9}}}\\{{\rm{1 - }}\sum\limits_{{\rm{k = 0}}}^{\rm{9}} {{{{\rm{(1 - p)}}}^{{\rm{k - 1}}}}} {\rm{p}}}&{{\rm{k = 10}}}\end{array}} \right.\end{array}\)

The expected value (or mean) is the sum of the product of each possibility \({\rm{x}}\) with its probability \({\rm{P(x)}}\)–

\(\begin{aligned}\mu &= \sum\limits_{{\rm{10}}} {\rm{x}} {\rm{P(x)}}\\&= \sum\limits_{{\rm{k = 1}}}^{\rm{9}} {\rm{k}} {\rm{P(X = k)}}\\&= \sum\limits_{{\rm{k = 1}}}^{\rm{9}} {\rm{k}} {{\rm{(1 - p)}}^{{\rm{k - 1}}}}{\rm{p + 10}}\left( {{\rm{1 - }}\sum\limits_{{\rm{k = 0}}}^{\rm{9}} {{{{\rm{(1 - p)}}}^{{\rm{k - 1}}}}} {\rm{p}}} \right)\\&= \sum\limits_{{\rm{k = 1}}}^{\rm{9}} {\rm{k}} {{\rm{(1 - p)}}^{{\rm{k - 1}}}}{\rm{p + 10 - 10}}\sum\limits_{{\rm{k = 0}}}^{\rm{9}} {{{{\rm{(1 - p)}}}^{{\rm{k - 1}}}}} {\rm{p Distributive Property}}\\&= 10 + (1 - 10)\sum\limits_{{\rm{k = 1}}}^{\rm{9}} {\rm{k}} {{\rm{(1 - p)}}^{{\rm{k - 1}}}}{\rm{p Combine like terms}}\\&= 10 - 9\sum\limits_{{\rm{k = 1}}}^{\rm{9}} {\rm{k}} {{\rm{(1 - p)}}^{{\rm{k - 1}}}}{\rm{p}}\end{aligned}\)

Therefore, the expression is obtained as \({\rm{10 - 9}}\sum\limits_{{\rm{k = 1}}}^{\rm{9}} {\rm{k}} {{\rm{(1 - p)}}^{{\rm{k - 1}}}}{\rm{p}}\).