What is the probability of floods in a year? The probability that one tornado will occur in a city in a year is $0.2$, the probability that two tornadoes will occur in a year is $0.01$, the probability that three or more tornadoes will occur is $0$.
The same city may have floods in a year that are caused by the thunderstorm that accompanies the tornado with probability $0.4$.
What is the probability of floods in a year?
My attempt
Define, $T_1$ the event that one tornado occurs, $T_2$ the event that two tornadoes will occur and $F$ the event of floods.
I will use the formula $$P(F)=P(F|T_1)P(T_1)+P(F|T_2)P(T_2)$$
Also I know that $$P(F|T_1)=\frac{P(F \cap T_1)}{P(T_1)} $$
and $$P(F|T_2)=\frac{P(F \cap T_1)}{P(T_2)} $$
It's given that $P(T_1)=0.2$,$P(T_2)=0.01$
But I am stuck at this point as I can't interpret the fact that the probability that floods due to tornadoes will occur is $0.4$, is it $P(F|T_1 \cup T_2)=0.4$?
Can you help?
 A: Let $T$ be the count for tornadoes (a random variable rather than an event). From the provided probabilities we have $T\in\{0,1,2\}$.
Well, you know that $\mathsf P(F\mid T\,{=}\,0)=0$ and $\mathsf P(F\mid T\,{=}\,1)=0.4$.   You should see that $\mathsf P(F\mid T\,{=}\,2)=1-(1-0.4)^2$ since this the probability that it is not the case that neither tornado results in a flood when there are two tornadoes.   So $\mathsf P(F\mid T\,{=}\,2)=0.64$.
Now just apply the Law of Total Probability:
$$\mathsf P(F) ~=~ \mathsf P(F\mid T\,{=}\,1)\,\mathsf P(T\,{=}\,1)+\mathsf P(F\mid T\,{=}\,2)\,\mathsf P(T\,{=}\,2)$$
A: The conditionals don't add up to the probability of $F$, but the joints do.
$$\begin{split}Pr(F)&=Pr(F, T_1)+Pr(F, T_2)\\
&=Pr(F|T_1)Pr(T_1)+Pr(F|T_2)Pr(T_2)\\
&=.4(.2)+.64(.01)=0.0864\end{split}$$
where $Pr(F|T_2)=.6(.4)+.4(.6)+.4^2$ (flood caused by first tornado but not second, flood caused by second tornado but not first, or flood caused by both tornados) or $1-.6^2$ (complement rule)
