Problem on conditional probability


#1

In my town, it’s rainy one third of the days. Given that it is rainy, there will be heavy traffic with probability 1/2, and given that it is not rainy, there will be heavy traffic with probability 1/4. If it’s rainy and there is heavy traffic, I arrive late for work with probability 1/2. On the other hand, the probability of being late is reduced to 1/8 if it is not rainy and there is no heavy traffic. In other situations (rainy and no traffic, not rainy and traffic) the probability of being late is 0.25. You pick a random day.

what is the probability that it is not raining and there is heavy traffic and i am not late?


#2

I will try to solve it with intuition

p(Rainy) = 1/3
p(heavy traffic when it is rainy) = 1/2 (so sample space is rainy),fav outcomes is heavy traffic
p(heavy traffic not rainy) = 1/4 (so sample space not rainy),fav outcomes is heavy traffic.
p(late when rainy and heavy traffic) = 1/2 (s0 sample space is rainy and heavy traffic)
p(late when not rainy and no heavy trafiic) = 1/8(so sample space not rainy and no traffic)
p(other) = .25 (sample space rainy no traffic and not rainy traffic)

p(not raining and heavy traffic and not late) = ? ,sample space is of not rainy and heavy traffic.

p(not rainy and heavy traffic ) = 1/4 so i thinnk answer is less than .25 (am i right ?)
how do i solve from here


#3

https://www.probabilitycourse.com/chapter1/1_4_5_solved3.php