Question 8.4: Felix states that he is able to distinguish a freshly ground......
Felix states that he is able to distinguish a freshly ground coffee blend from an ordinary supermarket coffee. One of his friends asks him to taste 10 cups of coffee and tell him which coffee he has tasted. Suppose that Felix has actually no clue about coffee and simply guesses the brand. What is the probability of at least 8 correct guesses?
Each guess is a Bernoulli experiment where the right answer is given with a probability of 50 %. The number of correct guesses therefore follows a binomial distribution, i.e. X ∼ B(10; 0.5). The probability of giving the right answer at least 8 times is identical to the probability of not being wrong more than 2 times. We can thus calculate P(X ≥ 8) as P(X ≤ 2):
P(X = 0) =\left(\begin{matrix} 10 \\ 0 \end{matrix} \right) 0.5^{0} \left(1 − 0.5\right) ^{10}\approx 0.000977
P(X = 1) = \left(\begin{matrix} 10 \\ 1 \end{matrix} \right) 0.5^{1} \left(1 − 0.5\right) ^{9}\approx 0.009766
P(X = 2) = \left(\begin{matrix} 10 \\ 2 \end{matrix} \right) 0.5^{2} \left(1 − 0.5\right) ^{8}\approx 0.043945.
This relates to
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
= 0.000977 + 0.009766 + 0.043945 ≈ 0.0547.
We would have obtained the same results in R using:
pbinom(2,10,0.5) 1-pbinom(7,10,0.5)