Resolved: Question 1 clarification

Hey Stephen,

Thank you for reaching out!

The multiplication rule states the following:

P(A∩B) = P(A|B) × P(B)

Mirroring the letters in the formula, we can also write it as follows:

P(B∩A) = P(B|A) × P(A)

However, P(A∩B) = P(B∩A). This is true since the intersection between two events is the same no matter whether we intersect A with B or B with A.

As an example of this, consider A being the even of drawing a Queen and B being the event of drawing a Spade. Intersecting event A with B results in the Queen of Spades. And intersecting event B with A also gives us the Queen of Spades.

Continuing this logic, we can write that 

P(A∩B) = P(B∩A)

and therefore that

P(A|B) × P(B) = P(B|A) × P(A)

This makes the following statement correct:

The probability of both events occurring simultaneously

P(B∩A)

equals the product of the likelihood of A occurring

P(A)

and the conditional probability that B occurs, given A has already occurred.

P(B|A)

Let me know if you need further assistance.

Kind regards,

365 Hristina