I came across this question recently and can't seem to find the correct approach.
Any help would be appreciated!
An experiment consists of first tossing an unbiased coin and then rolling a fair die.
If we perform this experiment successively, what is the probability of obtaining a heads on the coin before a $1$ or $2$ on the die?
$\mathbb P(\textrm{Heads})=\frac12$
$\mathbb P(1,2)=\frac13$
If $A_i$ represents the event that a $1$ or a $2$ is rolled on the $i^{th}$ toss, then I have to find the following:
$$\bigcup^{\infty}_{i=1}\mathbb P(A_i).$$
But I am not sure how to find this and also incorporate the probability of landing on heads before this…
Am I approaching this correctly or should I be assigning random variables and working from there?
Best Answer
A simple way to find the probability is to condition on the result of the first round. It is clear there is some probability $p$ of obtaining a head before (though not necessarily immediately before) a $1$ or $2$. Call that the probability of winning.
We win if (i) we get a head on the first round or (ii) we get a tail, don't roll a $1$ or $2$, but ultimately win.
The probability of (i) is $\frac{1}{2}$.
For (ii), note that the probability of tail and then something other than $1$ or $2$ is $\frac{1}{2}\cdot \frac{4}{6}$. Given this has happened, the probability of ultimately winning is $p$. Thus $$p=\frac{1}{2}+p\cdot \frac{1}{2}\cdot \frac{4}{6}.$$ Solve this linear equation for $p$. We get $p=\frac{3}{4}$.