Math Problem Statement
Suppose you just received a shipment of fifteen televisions. Four 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 of the two televisions does not work?
Solution
To solve this, we can break the problem into two parts. First, we compute the probability that both televisions work, and then we calculate the probability that at least one of the televisions does not work.
Part 1: Probability that both televisions work
There are 15 televisions in total, and 4 are defective. This means 11 of the televisions work. We need to calculate the probability that both selected televisions are not defective (i.e., they both work).
We will use combinations to solve this. The total number of ways to select 2 televisions from 15 is given by:
The number of ways to select 2 working televisions from the 11 that work is:
Thus, the probability that both selected televisions work is:
Part 2: Probability that at least one of the two televisions does not work
The probability that at least one of the selected televisions does not work is the complement of the probability that both televisions work. In other words:
Using the result from part 1:
Final Answers:
- The probability that both televisions work is .
- The probability that at least one of the two televisions does not work is .
Would you like more details on any part of the solution?
Here are five questions related to the problem:
- How do you calculate combinations and why are they used in this problem?
- What would the probability be if only one defective television were in the shipment?
- How does the probability change if more televisions are randomly selected?
- What is the difference between dependent and independent events in probability?
- Can you extend this approach to calculate probabilities for more complex systems?
Tip: For problems involving "at least one" outcomes, it is often easier to calculate the complement (opposite event) and subtract it from 1.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Probability
Combinatorics
Formulas
Combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)
Complement Rule: \( P(\text{at least one does not work}) = 1 - P(\text{both work}) \)
Theorems
Law of Total Probability
Combinatorial Counting
Suitable Grade Level
Grades 10-12
Related Recommendation
Probability of Sampling Defective Products from a Batch of 100
Probability of Accepting Shipment from 15 Modems with 3 Defective: Calculation and Explanation
Probability of Defective Generators in a Shipment of 16 Generators
Probability of Defective Components in a Pack of 10 Using Binomial Distribution
Probability of Rejecting a Shipment with Defective Components