Question

In a race, Brian Collins has to cross 10 hurdles. The probability that he clears a hurdle is . Find the probability.

P(at least three hurdles are cleared)

Answer 1

An important thing to recognize here is that he can only clear some number of hurdles each run.

This means that clearing 2 hurdles and clearing 3 hurdles are mutually exclusive events.

Thus:
\[
\Pr(X \geq 3) = \Pr(X = 3) + \Pr(X = 4) +\Pr(X = 5) + ...
\]

Answer 2

However, this is a cumbersome operation since we have to calculate 7 probabilities.

If we consider the probability of that he clears no more than three hurdles: \[
\Pr(X<3) = \Pr(X=0) + \Pr(X=1) + \Pr(X=2)
\]As you can see, this is much easier to calculate. The reason why we can do this is because we know:\[
\Pr(X \geq 3) + \Pr(X<3) = 1
\]
That is, the probability of an event happening or not happening is 1. Either has to happen.

Answer 3

@Rainbow_Dash Are you following?

Answer 4

yes, thank you very much :)