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Chapter 5: Problem 21

Suppose you just received a shipment of six televisions. Two of the televisions are defective. If two televisions are randomly selected, compute the probability that both televisions work. What is the probability at least one does not work?

Short Answer

Expert verified
The probability that both televisions work is \(\frac{2}{5}\). The probability that at least one does not work is \(\frac{3}{5}\).

Step by step solution

01

Determine Total Possible Outcomes

Calculate the total number of ways to select 2 televisions out of 6. This is given by the combination formula \(\binom{6}{2}\). The formula for combinations is \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\). Here, \(n = 6\) and \(r = 2\).
02

Calculate Total Possible Selections

Substitute the values into the formula: \(\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15\). There are 15 possible ways to select 2 televisions from 6.
03

Determine Favorable Outcomes for Both Working

Calculate the number of favorable outcomes where both televisions are working. We have 4 working televisions out of 6. The number of ways to choose 2 working televisions from these 4 is \(\binom{4}{2}\).
04

Calculate Favorable Selections

Substitute the values into the combination formula: \(\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6\). There are 6 ways to select 2 working televisions from 4.
05

Calculate Probability Both Working

The probability that both selected televisions work is the ratio of the favorable outcomes to the total possible outcomes: \(\frac{6}{15} = \frac{2}{5}\).
06

Calculate Probability At Least One Does Not Work

The probability that at least one television does not work is the complement of the probability that both work: \(1 - \frac{2}{5} = \frac{3}{5}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics
Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects. When solving problems like selecting televisions from a shipment, combinatorics provides the tools to determine how many ways we can choose certain items from a larger set. Understanding the basic concepts of combinatorics helps us navigate problems that involve combinations and permutations. In our problem, combinatorics helps us count the number of ways to select 2 televisions from 6.
Complement Rule
The complement rule is a fundamental probability principle that simplifies complex probability calculations. It states that the probability of an event occurring is equal to 1 minus the probability it does not occur. This is particularly useful when it is easier to calculate the probability of the complement event. Applying this rule in our problem, once we calculate the probability that both selected televisions work, using the complement rule helps us find the probability that at least one television does not work: \( P(A^c) = 1 - P(A)\).
Favorable Outcomes
In probability, favorable outcomes are the specific outcomes of interest within the set of all possible outcomes. These are the outcomes that satisfy the condition we are interested in. In our television selection problem, we are interested in the number of ways to select two televisions so that both work. To determine the favorable outcomes, we count the ways to choose two working televisions from the four available. Using the combination formula helps us easily find the favorable outcomes.
Combination Formula
The combination formula is used to determine the number of ways to choose a subset of items from a larger set without regard to the order of selection. The formula is given by: \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]. Here, \( n \) is the total number of items, and \( r \) is the number of items to choose. This formula is crucial in our problem for calculating both the total number of ways to select two televisions from six and the number of ways to select two working televisions from four. This helps us determine both the total and favorable outcomes required to calculate the desired probabilities.

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