A coffee connoisseur claims that he can distinguish between a cup of instant coffee and a cup of percolator coffee 75% of the time. It is agreed that his claim will be accepted if he correctly identifies at least 5 of the 6 cups. Find his chances of having the claim i) accepted, ii) rejected, when he does have the ability he claims.
Answers
Answer:
Step-by-step explanation:
The probability of his claim being accepted is 0.0195 or approximately 1.95%.
The probability of his claim being rejected is also 0.0195 or approximately 1.95%.
Let's assume that the coffee connoisseur does have the ability to distinguish between a cup of instant coffee and a cup of percolator coffee 75% of the time.
We can use the binomial distribution formula to calculate the probabilities.
i) To find the probability of his claim being accepted, we need to calculate the probability of him correctly identifying at least 5 of the 6 cups.
P(X >= 5) = 1 - P(X < 5)
where X is the number of cups he correctly identifies and P(X < 5) is the probability of him correctly identifying 4 or fewer cups.
P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
P(X < 5) = (6 choose 0) * (0.75)^0 * (0.25)^6 + (6 choose 1) * (0.75)^1 * (0.25)^5 + (6 choose 2) * (0.75)^2 * (0.25)^4 + (6 choose 3) * (0.75)^3 * (0.25)^3 + (6 choose 4) * (0.75)^4 * (0.25)^2
P(X < 5) = 0.9805
Therefore,
P(X >= 5) = 1 - P(X < 5) = 1 - 0.9805 = 0.0195.
So, the probability of his claim being accepted is 0.0195 or approximately 1.95%.
ii) To find the probability of his claim being rejected, we need to calculate the probability of him correctly identifying 4 or fewer cups.
P(X <= 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
P(X <= 4) = (6 choose 0) * (0.75)^0 * (0.25)^6 + (6 choose 1) * (0.75)^1 * (0.25)^5 + (6 choose 2) * (0.75)^2 * (0.25)^4 + (6 choose 3) * (0.75)^3 * (0.25)^3 + (6 choose 4) * (0.75)^4 * (0.25)^2
P(X <= 4) = 0.9805
Therefore,
P(X > 4) = 1 - P(X <= 4) = 1 - 0.9805 = 0.0195
So, the probability of his claim being rejected is also 0.0195 or approximately 1.95%.
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