Question

The Triangle Test A procedure often used to control the quality of name-brand food products utilizes a panel of five "tasters." Each member of the panel tastes three samples, two of which are from batches of the product known to have the desired taste and the other from the latest batch. Each taster selects the sample that is different from the other two. Assume that the latest batch does have the desired taste, and that there is no communication between the tasters. a. If the latest batch tastes the same as the other two batches, what is the probability that the taster picks it as the one that is different? b. What is the probability that exactly one of the tasters picks the latest batch as different? c. What is the probability that at least one of the tasters picks the latest batch as different?

Short answer

Answer: The probability is \(5 \cdot \frac{1}{3} \cdot \left(\frac{2}{3}\right)^{4}\).

Step-by-step solution

Determine the total number of choices.

The taster has three samples to choose from, so they have 3 choices.

Calculate the probability of picking the latest batch as different.

Since the taster has to pick one sample out of three, and all three samples taste the same, the probability of picking the latest batch as different is 1/3 because there is only one latest batch among the three samples. The required probability is \(\frac{1}{3}\). #b. Probability that exactly one of the tasters picks the latest batch as different.#

Possible outcomes.

In this case, we want exactly one taster out of the five to pick the latest batch as different. This can happen in multiple ways, for example, taster 1 picks the latest batch, and the rest do not, or taster 2 picks the latest batch, and the rest do not, and so on.

Use Binomial Probability.

We can use the binomial probability formula to determine the probability of exactly one taster picking the latest batch as different. The binomial probability formula is \(P(X=k) = \dbinom{n}{k} \cdot p^{k} \cdot (1-p)^{n-k}\), where \(P(X=k)\) is the probability of exactly \(k\) successes, \(n\) is the number of trials, and \(p\) is the probability of success in a single trial. In our case, we have \(n = 5\) trials (tasters), \(k = 1\) success (exactly one taster picking the latest batch), and \(p = \frac{1}{3}\) (probability of one taster picking the latest batch).

Calculate the probability of exactly one taster picking the latest batch as different.

Using the binomial probability formula: \(P(X=1) = \dbinom{5}{1} \cdot \left(\frac{1}{3}\right)^{1} \cdot \left(1-\frac{1}{3}\right)^{5-1} = 5 \cdot \frac{1}{3} \cdot \left(\frac{2}{3}\right)^{4}\). The required probability is \(5 \cdot \frac{1}{3} \cdot \left(\frac{2}{3}\right)^{4}\). #c. Probability that at least one of the tasters picks the latest batch as different.#

Calculate the probability of none of the tasters picking the latest batch as different.

First, we need to find the probability of none of the tasters picking the latest batch as different. Using the binomial probability formula with \(k = 0\): \(P(X=0) = \dbinom{5}{0} \cdot \left(\frac{1}{3}\right)^{0} \cdot \left(1-\frac{1}{3}\right)^{5-0} = \left(\frac{2}{3}\right)^{5}\).

Calculate the probability of at least one taster picking the latest batch as different.

Now, to find the probability of at least one taster picking the latest batch as different, we will subtract the probability of none of the tasters picking the latest batch as different from 1. Probability of at least one taster picking the latest batch as different is \(1 - P(X=0) = 1 - \left(\frac{2}{3}\right)^{5}\). The required probability is \(1 - \left(\frac{2}{3}\right)^{5}\).

Key concepts

Binomial Distribution

The binomial distribution is a fundamental concept in probability theory that models scenarios where there are a fixed number of independent trials, each of which results in a binary outcome: success or failure. In the context of quality control and statistical testing, the binomial distribution helps in determining the probability of a certain number of successes in these trials.

For example, in the scenario involving the panel of tasters, we treat each of the five tasters as an independent trial. Each taster can choose the latest batch as different (a success) or not (a failure).
The probability of success, in this case, is the probability of a taster picking the latest batch as different, which is \(\frac{1}{3}\).

To calculate specific probabilities, such as exactly one taster identifying the latest batch as different, we use the binomial probability formula:

• \(P(X=k) = \dbinom{n}{k} \cdot p^{k} \cdot (1-p)^{n-k}\)

where \(k\) is the number of successes (tasters who select the latest batch as different) among \(n\) trials.
This powerful formula allows us to carry out various statistical analyses to support decisions in quality control.

Quality Control

Quality control is a critical aspect of production processes, especially in industries like the food industry, where consistency in taste and quality can have substantial impacts.

An essential tool in quality control is the use of systematic testing to ensure that products meet specified standards. The triangle test, as described, is a practical mechanism used to evaluate the consistency of a food product's taste. By having a panel of tasters assess whether a new batch differs from known desired batches, producers can statistically assess product consistency.

• The triangle test relies on human perception, which introduces variability. Therefore, statistical methods, such as those based on binomial distribution, are employed to make sense of the results.
• This statistical approach helps determine probabilities related to tasters picking the latest batch as different either falsely or correctly.

In essence, quality control leverages the principles of probability and statistical testing to maintain product quality and consistency, ensuring that customer satisfaction remains high.

Statistical Testing

Statistical testing is a critical component of decision-making in quality control contexts. It involves using statistical methods to test hypotheses and determine the likelihood of different outcomes occurring by chance.

In the given exercise, statistical testing comes into play when evaluating probabilities, such as exactly or at least one of the tasters picking the latest batch as different. Particularly, we calculated and interpreted probabilities using:

• The probability of zero tasters selecting the latest batch (\(P(X=0)\)), which was computed as \(\left(\frac{2}{3}\right)^{5}\).
• The probability of at least one taster picking the latest batch, found by subtracting \(P(X=0)\) from 1, ensuring a complete understanding of the probability distribution.

Such statistical tests help in drawing conclusions about batch consistency and making informed decisions. This kind of analysis provides a measure of confidence about whether to approve or reject a batch, forming a cornerstone of quality assurance strategies in quality control.