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Probability is a fundamental concept in statistics and mathematics that measures the likelihood of an event occurring. In the context of this exercise, probability helps us determine how likely it is for your friend to correctly identify the three cups of tea where milk was added first if they are just guessing. To calculate probability, you use the ratio of the number of favorable outcomes to the total number of possible outcomes. Here, the favorable outcome is your friend correctly picking the specific three cups with milk added first. Since there is only one such specific combination,

• Number of favorable outcomes = 1
• Total number of possible outcomes = 20

Thus, the probability is calculated as:\[P(\text{success}) = \frac{1}{20}\]This tells us that there is a 5% chance your friend can successfully identify the three cups by pure guesswork.

Combinations refer to the selection of items where the order of selection does not matter. In this exercise, we are interested in finding out how many different combinations of three cups can be selected from a total of six cups. The formula for finding combinations is given by:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where:

• \( n \) is the total number of items,
• \( k \) is the number of items to choose, and
• \( ! \) denotes factorial, i.e., the product of all positive integers up to that number.

In this scenario, \( n \) is 6 (total cups) and \( k \) is 3 (cups you choose). So, we compute:\[\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20\]This means there are 20 different ways to choose any 3 cups from the 6 available without considering the order.

A blind taste test is a method to measure unbiased preferences by ensuring that the taster is unaware of the test conditions. This exercise involves using a blind taste test to challenge your friend’s claim about her ability to distinguish how the tea was made. In this setup:

• Your friend tastes all six cups without knowing which method was used in the preparation.
• The objective is to identify the three cups where milk was added first.

The purpose of a blind taste test like this one is to eliminate any preconceived notions or biases the taster might have about which method of preparing the tea tastes better or is more detectable. This ensures that any success in the test relies solely on taste and not on prior influence or information.