I am a bit confused, can you help?
So, let's say I have the task of selecting cards from a deck of playing cards. There are 52 cards total; 50% are red and 50% are black.
The probability of getting a red card is 50%
The probability of subsequently selecting a black card is 26/51 (because 1 card is removed)
But the probability of a red card, given we've already got a black card, I think is:
p(A∩B) = p(A) * p(B|A) – Which would be 0.5 * 0.51
But then I hear about a different formula for conditional probabiltiy, where I should divide the result by the probability of selecting a black.
I am confused, as I watched a youTube video which explained that the multiplication rule works like I have it & then read some blogs about dividing by the p(b).
Can someone help me figure out why we do that & whether my initial view was correct?
Thanks a lot
Best Answer
"...the probability of a red card, given we've already got a black card, is $P(A\cap B)=\dots$."
No. $P(A\cap B)$ is the probability of getting a red card and then getting a black card (or getting a black card and then getting a red card). The probability of getting a red card, given we've already got a black card is a conditional probability, denoted $P(A\mid B)$, and calculated by $$P(A\mid B)={P(A\cap B)\over P(B)}$$ In your example, $P(A\cap B)=(26/52)(26/51)=13/51$, while $P(A\mid B)=P(A\cap B)/P(B)=(13/51)/(26/52)=26/51$.