Confusion about Joint Probability Scenario 1:
From a deck of 52 cards, the joint probability of picking up a card that is both red and 6 is
By Counting : 2/52 = 1/26
By using multiplication rule : P(6)×P(red)= (4/52 × 26/52) = 1/26
Scenario 2:




Planned to Travel
Travelled
Did Not Travel
Total




Yes
200
50
250


No
100
650
750


Total
300
700
1000




The Joint Probability of people that plan to travel and actually travelled is 200/1000 = 0.2
Question:
Why is the Joint Probability of people that plan to travel and actually travelled is not
P(Plan to travel) x p(Travelled) = 250/1000 x 300/1000 = 75/1000
Can some math guru please explain me.

The joint probability of event $A$ and $B$ is defined as
$$
P(A,B) = P(A|B)P(B) = P(B|A)P(A).
$$
This could be understood intuitively as the occurrence of either $A$ or $B$ might influence the probability of the other happening (called dependent events). Scenario 1 is a special case, in which the occurrence of either event does not affect the probability of the other (A and B are independent events), i.e. there are the same number of 6s that are red compared to those that are in other individual colors, so
$$
P(A|B) = P(A) \Rightarrow P(A,B) = P(A)P(B).
$$
But the same is not true for the second scenario. There is dependency in the data such as there are more people in the “planned to travel” category that travelled. So to calculate the joint probability mathematically, we have to multiply P(Planned to travel given travelled) by P(travelled). The formula for the conditional probability is given by
$$
P(A|B) = \frac{P(A,B)}{P(B)}
$$
which already has $P(A,B)$ in the numerator, so in this type of scenario, the joint probability is rarely calculated multiplicatively.
