Seven terminals of an interactive system at Crocker Ship are...

Seven terminals of an interactive system at Crocker Ship are attached by a communication line to the central computer. Exactly four of the seven terminals are ready to transmit a message. Assume that each terminal is equally likely to be in the ready state. Let $X$ be the random variable whose value is the number of terminals polled until the first ready terminal is located. (a) What values may $X$ assume? (b) What is the probability that $X$ will assume each of these values? Assume that terminals are polled in a fixed sequence without repetition. (c) Suppose the communication line has $m$ terminals attached, of which $n$ are ready to transmit where $n \geq 1$. Show that $X$ can assume only the values $i=1,2, \ldots, m-n+1$ with $p(i)=P[X=$ $i]=\left(\begin{array}{c}m-i \\ n-1\end{array}\right) /\left(\begin{array}{c}m \\ n\end{array}\right)$.

Seven terminals of an interactive system at Crocker Ship are attached by a communication line to the central computer. Exactly four of the seven terminals are ready to transmit a message. Assume that each terminal is equally likely to be in the ready state. Let $X$ be the random variable whose value is the number of terminals polled until the first ready terminal is located.
(a) What values may $X$ assume?
(b) What is the probability that $X$ will assume each of these values? Assume that terminals are polled in a fixed sequence without repetition.
(c) Suppose the communication line has $m$ terminals attached, of which $n$ are ready to transmit where $n \geq 1$. Show that $X$ can assume only the values $i=1,2, \ldots, m-n+1$ with $p(i)=P[X=$ $i]=\left(\begin{array}{c}m-i \\ n-1\end{array}\right) /\left(\begin{array}{c}m \\ n\end{array}\right)$.

Key Concepts

Sampling Without Replacement

Sampling without replacement means that once an element (such as a terminal) is selected, it cannot be selected again, which alters the probabilities of subsequent selections. This concept is important in the given problem, as the polling process depletes the set of terminals sequentially, affecting the chances of finding a ready terminal as the polling progresses.

Hypergeometric Distribution

The hypergeometric distribution applies to scenarios where sampling is done without replacement from a finite population that contains a fixed number of successes and failures. It is relevant here because the terminals are polled in a fixed order without repetition, and the probability calculations depend on the number of ready terminals present among a set total.

Combinatorial Analysis

Combinatorial analysis involves counting the number of ways in which events can occur, using tools such as combinations and binomial coefficients. This analysis is crucial in probability problems where outcomes depend on arrangements and selections, such as determining the number of favorable polling sequences against the total possible sequences.

Discrete Random Variable

A discrete random variable is one that takes on a countable number of distinct values. In problems involving sequential polling or waiting times, the random variable is defined to represent the number of trials needed to achieve a certain outcome, such as the first success, making it essential in determining probabilistic outcomes.

Support of a Random Variable

The support of a random variable is the set of all values that the variable can assume with nonzero probability. In the context of polling terminals, identifying the possible positions at which the first success can occur defines the support, ensuring that the derived probability mass function is properly defined for all feasible outcomes.

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