Question

Thirty percent of the CFA candidates have a degree in economics. A random sample of three CFA candidates is selected. What is the probability that at least one of them has a degree in economics?

Answer 1

This is an example of a random variable that follows Bernoulli distribution.

Whenever you perform $n$ experiments, with probability $p$ of success, the provability of havin $k$successes is

$P(X=k)=\displaystyle\binom{n}{k}p^k(1-p)^{n-k}$

In this case, you have $n=3$ (you choose a sample of 3 candidates), $p=0.3$ (30% of candidates have a degree in economics) and we want $k$ to be at least one.

One way to solve this is to consider that

$P(X\geq 1)=P(X=1)+P(X=2)+P(X=3)$

But it is much quicker to observe that

$P(X\geq 1)=1-P(X$

And we have

$P(X=0)=\displaystyle\binom{3}{0}0.3^0\cdot 0.7^{3}=1\cdot 1\cdot 0.7^3$

So, the probability that at least one of them has a degree in economics is

$1-0.7^3 = 0.657$