Monday puzzle: lady tasting tea problem

In a previous post, I discussed ways to design a better blind taste test.

This puzzle is about a taste test with historic significance to statistics.

The problem is based on an incident at a 1920s tea party in Cambridge. The story goes one lady claimed the ability to distinguish between tea made by pouring tea to milk or by adding milk to the tea.

Everyone questioned the claim, but one person decided test it out. He created an experiment with 8 tea cups, consisting of 4 cups of each preparation.

The lady was remarkably able to identify all 8 cups, raising the issue of whether she just got lucky.

What are the odds that the lady identified all 8 cups by chance?

The problem is known as the Lady Tasting Tea, and it brought about the more modern analysis of testing random experimental data.

(I want to mention a related story of my own. When I was about 10 years old, one of my friends claimed he could tell the difference whether cereal was made by pouring milk over the cereal or vice versa. I highly doubted he could tell the difference. If only I had known this story back then…)

Can you solve it? The answer is below.
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The answer

This is a classic question of combinations. We know there are 8 items, of which 4 will be one type, and 4 the other.

Therefore there are “8 choose 4″ or 70 different combinations that are possible. The probability of identifying all of the cups by chance is a mere 1/70 (around 1.4 percent). We can conclude the lady likely had a refined palette.