Probability / Conditional Independence (Wine Tasting)

Question

I've been studying probability and am struggling a bit with the following problem:

There are three amateur wine tasters -- Jane, Bob and Shelley -- who classify wine as either "sweet" or "not sweet" after blind tasting. Assume that each of them gets it right 75% of the time, and that the classifications are independent conditioned on the label.

Suppose we now get a fourth person, Charlie. Charlie classifies wine not by tasting it himself, but by simply following the majority decision of our 3 wine tasters (e.g. if 2 or more of the wine tasters classify a given wine as sweet, Charlie will also classify it as sweet, and vice versa).

1) What is the probability that Charlie will classify a given wine correctly? (If insufficient information, explain)

2) Now suppose that the "independent, conditioned on the label" assumption mentioned above does not hold for our three wine tasters. What is the lower bound on the probability that Charlie will classify a given wine correctly? (If insufficient information, explain)

For (1), I made some progress, though I'm not totally sure what is meant by "independent, conditioned on the label". Does it just mean that how one taster classifies it has no effect on how another taster classifies it? If so, there are 4 ways in which the majority get it right:

a) All three get it right -> (.75)(.75)(.75) = 0.421875
b) Either Jane, Bob, or Shelley get it wrong, and the other two get it right -> (.75)(.75)(.25) = 0.146025

So since Charlie always goes with majority decision...
Pr(Charlie gets it right) = 0.421875 + 0.146025 + 0.146025 + 0.146025 = 0.84375. Is this right?

For (2), I'm confused as to how to tackle this. If the independence assumption doesn't hold, does this imply that, for example if Jane classifies given wine as sweet then Bob and Shelley are more likely to classify it as sweet as well? But the problem doesn't really say what the precise nature of the dependence is...isn't it also possible that Bob and Shelley would be more likely to classify it in the opposite manner because they don't like Jane or whatever?

So I'm just not sure how to compute this lower bound, or whether there is insufficient information to answer this either way. Thanks for the help.

Answer 1

It looks like you are right!

Part 1

Let $ X $ denote the number of people (out of Jane, Bob, and Shelley) who correctly classify the wine. Then $ X \sim Binomial(n=3,p=0.75) $. Since Charlie follows the majority, he will correctly classify the wine if $ X \geq 2 $.

$$ P(X \geq 2) = P(X=2) + P(X=3) = {{3}\choose{2}}(0.75)^2(0.25)^1 + {{3}\choose{3}}(0.75)^3(0.25)^0 = 0.8438 $$

Part 2

In order to calculate the probability that Charlie correctly classifies the wine, you once again need to calculate $ P(X\geq2) $. However, since the wine tasting events are no longer independent, you need more information in order to explain this dependence. There is no way to answer this question without that information.