A mail-order computer business has six telephone lines. Let...

A mail-order computer business has six telephone lines. Let $X$ denote the number of lines in use at a specified time. Suppose the pmf of $X$ is as given in the accompanying table. Calculate the probability of each of the following events. (a) $\{$ at most three lines are in use $\}$ (b) $\{$ fewer than three lines are in use $\}$ (c) $\{$ at least three lines are in use $\}$ (d) $\{$ between two and five lines, inclusive, are in use $\}$ (e) $\{$ between two and four lines, inclusive, are not in use $\}$ (f) $\{$ at least four lines are not in use $\}$

A mail-order computer business has six telephone lines. Let $X$ denote the number of lines in use at a specified time. Suppose the pmf of $X$ is as given in the accompanying table.
Calculate the probability of each of the following events.
(a) $\{$ at most three lines are in use $\}$
(b) $\{$ fewer than three lines are in use $\}$
(c) $\{$ at least three lines are in use $\}$
(d) $\{$ between two and five lines, inclusive, are in use $\}$
(e) $\{$ between two and four lines, inclusive, are not in use $\}$
(f) $\{$ at least four lines are not in use $\}$

Key Concepts

Summation of Probabilities

In problems involving discrete random variables, calculating the probability of a complex event often involves summing up the probabilities of individual outcomes as specified by the pmf. This aggregation method is essential when the event is defined over a range of values, and each probability contributes to the total probability of the event.

Complement Rule

The complement rule is a fundamental concept in probability that states the probability of an event occurring is one minus the probability that it does not occur. This technique is particularly useful when calculating the probability of an event is simpler by subtracting the probability of its complement from one, such as computing probabilities for inequalities by considering the non-occurrence of the complementary event.

Probability Mass Function (pmf)

The probability mass function (pmf) gives the probability that a discrete random variable is exactly equal to some value. It is fundamental in calculating event probabilities by summing up the probabilities assigned to the relevant outcomes. In many real-world problems, the pmf allows for determining the likelihood of each possible state of the system.

Discrete Random Variable

A discrete random variable is one that can take on a countable number of distinct values. In probability theory, it is used to model scenarios where outcomes are separate and distinct, such as the number of telephone lines in use. Each possible value the variable can assume is associated with a specific probability, and these probabilities must sum to one over the entire sample space.

Events and Inequalities in Probability

Events in probability are collections of outcomes, and they can be defined using inequalities such as 'at most', 'at least', 'fewer than', and 'between'. Understanding these terms translates to setting the correct limits for summing probabilities. For example, 'at most three' includes outcomes 0, 1, 2, and 3, while 'at least three' includes outcomes 3, 4, and so on.

Transcript

00:01 Okay, so here we have the data given in the table above, and for part a, we want to find the probability that x is less than or equal to 3, because we want to see what is the probability that at most three lines are in use.

00:18 So that's given by probability of x less than equal to 3, which can be written as, probability of x equals 0, all the way to 3, and that can be summed using the values in the table.

00:35 Which is 0 .7.

00:38 Now for part b we want to figure out what's the probability that there are fewer than three lines in use so that's x less than 3 which is everything except this term here so that comes out to 0 .45 and for part c at least three lines are in use so x greater than are equal to 3 which can be written in a different way which is one minus so we're finding the complementary probability for x less than 3 so that's 1 minus 0 .45 which is 0 .55 and for part d between 2 and 5 lines inclusive are in use so what does that mean that means that we're including both endpoints both 2 and 5 so we have less than 2 and less than equal to here so this is 1 minus probability of x equals 1 and x equals 6 and also x equals 0 so this is 1 minus 0 .15 minus 0 .04 minus 0 .1 which is 0 .7 and for part e we want to figure out the probability that 2 and 4 lines inclusive are not in use okay so what does that mean between 2 and 4 lines are not in use that means we can write it in this form.

02:28 We can say that 2 is less than equal to the complementary quantity to x, which is 6 minus x.

02:41 So instead of writing 2 less than equal to x less than or equal to 4, we're writing 2 less than equal to 6 minus x less than or equal to 4 because x is the number of lines in use.

02:52 So this is the number of lines not in use.

02:54 So this is equal to probability of negative 4, less than equal to minus x, less than equal to negative 2.

03:07 Or in other words, by multiplying the inequality by negative 1, all across, we have 2 less than equal to x, less than or equal to 4...

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