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Chapter 3: Problem 55

Twenty percent of all telephones of a certain type are submitted for service while under warranty. Of these, \(60 \%\) can be repaired, whereas the other \(40 \%\) must be replaced with new units. If a company purchases ten of these telephones, what is the probability that exactly two will end up being replaced under warranty?

Short Answer

Expert verified
The probability is approximately 14.6%.

Step by step solution

01

Define the Problem

We need to find the probability that exactly two telephones out of ten will be replaced under warranty. We know that 20% of the telephones are submitted for service and out of those, 40% must be replaced.
02

Determine Key Probabilities

Calculate the probability that a single telephone is submitted for service and then replaced. - Submitting for service: 20% or 0.2 - Replacing if submitted: 40% or 0.4 Thus, the probability of being replaced = \(0.2 \times 0.4 = 0.08\).
03

Apply Binomial Probability Formula

Since we need the probability that exactly two out of ten are replaced, we use the binomial probability formula:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \(n = 10\), \(k = 2\), \(p = 0.08\).
04

Calculate the Binomial Coefficient

The binomial coefficient \(\binom{10}{2}\) calculates how many ways we can choose 2 telephones to be replaced out of 10.\[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = 45 \]
05

Compute Probability for Exactly 2 Replacements

Substitute the values into the binomial formula:\[ P(X = 2) = \binom{10}{2} (0.08)^2 (0.92)^8 \]Calculate further:\[ P(X = 2) = 45 \times (0.08)^2 \times (0.92)^8 \]\[ P(X = 2) = 45 \times 0.0064 \times 0.5132 \]\[ P(X = 2) \approx 0.146 \]
06

Interpretation

The probability that exactly two out of the ten telephones will be replaced under warranty is approximately 14.6%.

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Key Concepts

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

Probability Theory
Probability theory forms the foundation of statistical analysis and decision making in uncertain scenarios. It involves calculating the likelihood of various possible outcomes, facilitating informed decisions and predictions.
For the given exercise, probability theory helps to determine the chance that a specific number of telephones will need replacement among the ten purchased. We begin by identifying the relevant events, such as submission for service and eventual replacement. Each event has a calculated probability impacting the final result.
By multiplying the probability of a single event being replaced (computed as 0.08) with its occurrence through the formula for binomial probability, a deep understanding of probability theory allows us to compute precise likelihoods for practical queries.
Binomial Coefficient
The binomial coefficient is a crucial element in probability and combinatorial calculations. It represents the number of ways to choose a subset of items from a larger set, commonly denoted as \( \binom{n}{k} \).
In our problem, we aim to find all possible combinations of choosing exactly two telephones for replacement out of the ten. The binomial coefficient, calculated as \( \binom{10}{2} = 45 \), tells us this number as 45.
This concept facilitates solving probability problems where specific outcomes need enumeration, ensuring precise computation in complex situations involving multiple possibilities.
Statistical Analysis
Statistical analysis involves interpreting data comprehensively for insights and decisions. It includes using mathematical tools like probability and distributions to analyze data trends and outcomes.
In this scenario, statistical analysis was used to calculate the probability of specific outcomes using the binomial distribution. The probability distribution is instrumental in predicting various outcomes based on observed probabilities鈥攕uch as the likelihood of two specific telephones needing replacement.
By applying statistical analysis, we translate raw probabilities into practical conclusions, like knowing the chance of two units being replaced is approximately 14.6%. This process is central to data-driven decision making in real-world applications.

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Most popular questions from this chapter

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