In this example, If temp = high then there is chance for weather being sunny is 60%. And P(temp = high) = 30%. The highlighted box represents days with high temperature and weather being sunny. In other words, P(temp=high ∩ weather = sunny) = 60% * 30% = 18% As far as I know we multiply to get joint probability. So here I am trying to find intersection which is joint probability and I am doing multiplication operation.
Now, My question is, Why we need to do multiplication here? From kindergarten multiplication has been known as A*B. where A is added B times to itself. Which clearly does not signify the relationship of intersection. What I am missing here?
Well, it is just by the definition of conditional probability that:
$$\mathsf P(\text{high_temp}\cap\text{sunny})=\mathsf P(\text{high_temp})\cdot\mathsf P(\text{sunny}\mid\text{high_temp})$$
Or in terms of generic events $A,B$:$$\mathsf P(A\cap B)=\mathsf P(A)\cdot\mathsf P(B\mid A)$$
Or with a little rearrangement: $\mathsf P(B\mid A)=\dfrac{\mathsf P(A\cap B)}{\mathsf P(A)}$
That is, the conditional probability of $B$ when given $A$ is just the measure of the probability for the intersection ($A\cap B$) relative to that for the given event ($A$).