# What is the probability that the “smartest” of them has IQ score above 120?

Suppose that we model the distribution of IQ scores in the general population as a normal random variable with mean 100 and standard deviation 15. Find the probability that a randomly selected person's IQ score is between 125 and 130.

Suppose that 5 people are chosen independently at random. What is the probability that the “smartest” of them has IQ score above 120?

How do account for the "smartest" of them? Where do I start?

• "the smartest of them has IQ score above 120" is the opposite of "all of them have IQ score below 120" – miracle173 Mar 21 '14 at 1:06

The first question is simply applying the normal distribution.

For the second problem: $\Pr(\max_{i \le n \le 5} X_i \le 120) = \Pr(X_i \le 120)^5$. Now use $\Pr(\max_{i \le n \le 5} X_i > 120) = 1 - \Pr(\max_{i \le n \le 5} X_i \le 120)$.

Let $X_i$ be $i$'s IQ. This is a random variable such as $X = 100 + 15G$ where $G\sim N(0,1)$. Now assume that random selected person's IQs are independant:

1. $P(\text{a randomly selected person's IQ score is between 125 and 130})=P_1$ $$P_1 = P(125<100 + 15G<130) = P(5/3 <G<2)$$
2. Assume that "more intelligent" means "with highest IQ" (hum.). $P(\text{the “smartest” of 5 people chosen independently...})=P_2$. $$P_2 = P(\max_{1\le i\le 5} X_i\ge 120) = 1 -P(\max_{1\le i\le 5} X_i<120) \\= 1 -P( X<120)^5 = 1 - P(G < 4/3)^5$$ We use independance in the third equality.