Question

A single person is hired to taste and rank each of three brands of tea, which are unmarked except for identifying symbols \(A, B,\) and \(C .\) If the taster has no ability to distinguish a difference in taste among teas, what is the probability that the taster will rank tea type \(A\) as the most desirable? As the least desirable?

Short answer

Answer: The probability of tea type A being ranked as the most desirable is \(\frac{1}{3}\), and the probability of it being ranked as the least desirable is also \(\frac{1}{3}\).

Step-by-step solution

Total number of permutations

There are 3 types of tea, so there are 3! (factorial) possible ways to rank the teas. Let's calculate 3!: 3! = 3 × 2 × 1 = 6. So, there are 6 total possible ways the person can rank the tea types.

Probability of A being the most desirable

To find the probability that A is ranked as the most desirable, we multiply its probability of being first by the possibilities of the other teas. Since the person can't distinguish, any type of tea has a \(\frac{1}{3}\) probability of being ranked first: Probability of A being first = \(\frac{1}{3}\). Now, let's think about the other two tea types. If tea A is ranked first, then the person has to choose between the remaining two teas, B and C. So, there are 2! possible ways to rank B and C: 2! = 2 × 1 = 2. Therefore, there are 2 favorable ways for tea A to be the most desirable. Now, to calculate the probability, we divide the number of favorable ways (2) by the total number of possible ways (6): Probability of A being the most desirable = \(\dfrac{2}{6}\) = \(\dfrac{1}{3}\).

Probability of A being the least desirable

To find the probability that A is ranked as the least desirable, we have to think about how many favorable ways can A be ranked last. If A is the least desirable, then the person has to choose between the remaining two teas, B and C for the first position. So, there are 2 possible ways to rank B and C in the first two positions. Therefore, there are 2 favorable ways for tea A to be the least desirable. Now, to calculate the probability, we divide the number of favorable ways (2) by the total number of possible ways (6): Probability of A being the least desirable = \(\dfrac{2}{6}\) = \(\dfrac{1}{3}\). So, the probability of tea type A being the most desirable is \(\dfrac{1}{3}\), and the probability of it being the least desirable is also \(\dfrac{1}{3}\).

Key concepts

Permutations and Combinations

Understanding permutations and combinations is critical when solving problems related to the arrangement of objects, like the ranking of different teas. Permutations are arrangements where the order matters, while combinations are selections where the order does not matter. For instance, ranking three teas in order (A, B, C) is a permutation because switching the order (C, B, A) results in a different ranking.

In the tea tasting problem, the person ranks three types of teas, and since order matters here (the ranking is specific), we are dealing with permutations. The number of permutations of 'n' items is calculated using factorial notation, which brings us to the factorial concept.

In terms of improving the exercise, emphasizing the difference between situations where permutations are used (order important) versus combinations (order not important) can significantly enhance understanding and application of these concepts.

Factorial Notation

The factorial of a number, represented by an exclamation point (!), is the product of all positive integers less than or equal to that number. For example, for number 3, we denote its factorial as 3! and calculate it as:
3! = 3 \(\times\) 2 \(\times\) 1 = 6.

Factorials are extremely useful in calculating permutations as they give us the total number of ways 'n' objects can be arranged or ordered. In the exercise, 3! represents the total permutations for the rank order of three teas. Knowing how to calculate and use factorial notation is the backbone for solving many problems in probability.

Points to reiterate for exercise improvement include the mechanical process of computing factorial notation and its practical significance in permutations.

Probability Calculation

Probability calculation is fundamental in determining how likely an event is to occur. The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. The formula to calculate the probability (P) is:
P = \(\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\).

In our tea tasting problem, the probability of tea A being ranked as the most and least desirable both turned out to be \(\dfrac{1}{3}\), as there were 2 favorable outcomes for each scenario out of 6 possible outcomes. Improving the exercise entails ensuring that students not only plug numbers into the formula but also understand the logic behind identifying favorable and possible outcomes, especially in contexts where it's not immediately clear, such as events with equal chances, like an inability to distinguish tastes.

Using real-life examples and explaining events that have equal probability may assist learners in grasping these probability calculations more intuitively.