Problem 53 - Basic Statistics for Business and Economics with Student

Question

The National Aeronautics and Space Administration (NASA) has experienced two disasters. The Challenger exploded over the Atlantic Ocean in 1986 and the Columbia exploded over East Texas in \(2003 .\) There have been a total of 113 space missions. Assume failures continue to occur at the same rate and consider the next 23 missions. What is the probability of exactly two failures? What is the probability of no failures?

Step-by-Step Solution

Answer

The probability of exactly two failures is approximately 0.0424, and no failures is approximately 0.6460.

Step 1: Understand the Problem

We are told there have been 2 failures in 113 missions, and we want to find the probability of 2 failures and 0 failures in the next 23 missions, assuming failures continue at the same rate.

Step 2: Determine the Failure Rate

The failure rate from past missions is calculated as the number of failures divided by the total number of missions. Hence, the failure rate (\(p\)) is \( \frac{2}{113} \).

Step 3: Define the Binomial Probability Formula

For a binomial distribution, the probability of getting exactly \(k\) failures in \(n\) trials is given by \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( \binom{n}{k} \) is the binomial coefficient.

Step 4: Calculate Probability of 2 Failures

Using \(n = 23\), \(k = 2\), and \(p = \frac{2}{113} \), calculate \( P(X = 2) = \binom{23}{2} \left(\frac{2}{113}\right)^2 \left(1 - \frac{2}{113}\right)^{21} \). Compute this value using these parameters to find the probability of exactly 2 failures.

Step 5: Calculate Probability of 0 Failures

Use the same formula with \(k = 0\). The probability of 0 failures is \(P(X = 0) = \binom{23}{0} \left(\frac{2}{113}\right)^0 \left(1 - \frac{2}{113}\right)^{23} \). Compute this probability to find the result for no failures.

Key Concepts

Probability Calculation

In statistics, calculating the probability of certain outcomes is critical to predicting future events based on past experiences, just like in NASA's space missions example. For these calculations, we rely on probability distribution models. Each outcome like having exactly 2 failures or no failures in space missions is subject to a probability calculation.

To determine the probability of an event happening, we need the failure rate (or success rate, depending on context) from past data. For a given number of trials, like 23 space missions, we use this rate to predict potential outcomes. Calculating the chance of exactly two failures, we use the binomial probability formula. This formula requires the binomial coefficient and the failure rate raised to the power of the number of desired events.

• Initial Data: past 113 missions, with 2 failures.
• Calculated Failure Rate: \(\frac{2}{113}\).
• Probability Formula: Combines past failure rate with desired new outcome trials and outcomes.

Understanding this calculation helps in decision-making, offering insights into future risks or success rates.

Failure Rate

Failure rate is a concept that explains how often failures happen in a given context. In situations like space missions, monitoring and calculating failure rates helps organizations like NASA predict and manage risks effectively. It's essentially the chance of failure occurring in one single instance, based on historical data.

For NASA, the failure rate is derived from past mission success and failure data: 2 failures out of 113 missions gives a failure rate expressed as a fraction, \(\frac{2}{113}\). This fraction tells us how likely a failure was historically and is useful for estimating future risks.

• Essentially a probability of failure in a single trial (one mission).
• Used as an empirical foundation for future mission planning.

With this rate, NASA, or any other organization, can strategize on enhancing safety measures, aiming to lower the failure rate over time.

Binomial Coefficient

The binomial coefficient is a crucial part of calculating probabilities in scenarios with two possible outcomes: success or failure. When dealing with a binomial distribution, like figuring out the probability of exactly two failures in space missions, the binomial coefficient helps determine how many combinations there can be.

Mathematically, the binomial coefficient \(\binom{n}{k}\) denotes how many ways \(k\) successes (or failures) can occur in \ trials. It is calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). In our space mission example:

• \(n\) represents total missions (23).
• \(k\) signifies the exact number of failures (e.g., 2).

The value of \(\binom{23}{2}\) gives us the number of ways to arrange two failures within 23 missions. This coefficient, combined with probabilities, assists in finding the likelihood of different specific outcomes.