A car race is very dangerous, and a crash can cause serious injuries. The league requires that anyone who has a crash to have a medical examination before they are allowed to race again. A certain racer has an independent .04 probability of a crash in a race
a) What is the probability that he will have his first crash within the first 30 races she runs this season?
My solution
Treating like the lottery problem
Probability of not having a crash
P(First Crash In 30 Races) = $.04^{30}$
But this produces the wrong answer since the book says that the answer is $.7061$.
Could someone explain why my approach doesn't work and how to properly go about this problem?
What you're calculating is the probability that he will crash in every race in the first $30$ races. The probability of a crash in a single race is $.04$, and so $.04^{30}$ represents the probability of getting a crash in every single of the first $30$ races.
Instead, the way to approach this problem is to note that the probability of not crashing in a race is $1 - .04 = 0.96$. To calculate the probability that the racer will have his first crash within the first $30$ races is the same as $1$ minus the probability that he won't have a single crash in the first $30$ races. Do you know where to go from here?