Every year, you observe the performance of 100 different mutual funds. Specifica

Question

Every year, you observe the performance of 100 different mutual funds. Specifically, you observe whether they perform better than the S&P500 or worse. If the mutual fund performance is determined entirely by luck, rather than skill of the manager, each mutual fund has only a 50% probability of beating the S&P 500 each year.

Suppose you are not sure whether skill affects mutual fund performance or not. You notice that five of the 100 funds have beaten the market every year for the last five years. What is the probability of observing at least that many funds beating the market every year for five years, if performance is determined entirely by luck?

Explanation / Answer

This is a binomial probabiliyt problem as the probability of luck in aiding the mutual fund is yes or no.

P(luck) = 0.5

totally 100 funds are there.

Let x be the number of funds beating the market by luck.

Then the probability of atleast 5 out of 100 funds beating the market by luck is

P( X >= 5) = 1- P(X<5)

= 1-(P (X=0)+P (X=1)+P (X=2)+P (X=3)+P (X=4))

P(X=x) = nCx p^x (1-p)^(n-x)

P (X=0) = 100C0 0.5^0 (1-0.5)^(100-0)

So the probability is almost 1.

X nCxp^x (1-p)^(n-x) 0 7.88861E-31 1 7.88861E-29 2 3.90486E-27 3 1.27559E-25 4 3.0933E-24 3.22484E-24 1

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