One half of the customers of an insurance company are women, the other half are men. Each year the probability that a man has a car accident is $\beta$. Also, each year the probability that a woman has a car accident is $\alpha$. The probability that an arbitrary customer has an accident this year is $\frac{\alpha+\beta}{2}$.
What is the probability that a randomly chosen customer doesn't have two accidents in two consecutive years?
To be clear, we are looking for the probability of the events:
year $i$ accident and year $i+1$ no accident
year $i$ no accident and year $i+1$ accident
year $i$ no accident and year $i+1$ no accident
Let be $A_i$ the event that an accident occurs in year $i$ and $F$ the event that a chosen customer is female. We know from the information that $P(A_i|F)=\alpha$ and $P(A_i)=\frac{\alpha+\beta}{2}$. We are looking for the probability $$1-P(A_{i+1}\cap A_i\mid F)-P(A_{i+1}\cap A_i\mid F^c).$$ If I check $P(A_{i+1}\cap A_i\mid F)$ I got stuck after some manipulations (of course the same happens with $P(A_{i+1}\cap A_i\mid F^c)$): $$ \begin{align*} &P(A_{i+1}\cap A_i\mid F)=\frac{P(A_{i+1}\cap A_i\cap F)}{P(F)}\frac{P(A_i\cap F)}{P(A_i\cap F)}\\ &=P(A_{i+1}\mid A_i\cap F)P(A_i\mid F)=P(A_{i+1}\mid A_i\cap F)\alpha=\dots? \end{align*}$$ We have no information about $P(A_{i+1}\mid A_i\cap F)$. Am I missing something? It seems to me that this problem can't be solved without further information!?