I have this question that I'm trying to solve:
Let's say we have two dice: one is blue and the other one is red. You roll the dices and try to predict the
outcome. Let's say the outcome of the blue die is b, and the outcome of the red is r.
You can understand that the outcome of the dices are independent of each other.
D = `sum of the outcomes (b+r) is an even number '
What is the probability of b = 3 and r = 5, which is P(b=3,r=5 | D) ?
The Answer that I have tried:
P (b =3, r =5/ D) = P(b= 3/ r= 5, D) * P( r = 5, D) = ⅓ * ⅙ = 1/18
Is my reasoning correct? or am I missing something?
To me you should just go directly from the multiplication rule of probability, that $P(A\cap B)=P(A|B)*P(B)$, converting it to the form $P(A|B)=\frac {P(A\cap B) } {P(B)}$
$P(blue=3\cap red=5|sum\ is\ even)=\frac {P(blue=3\cap red=5) } {P(sum\ is\ even)}=\frac {1/36} {18/36}=\frac 1 {18}$
Another way to do it just by thinking about it is to realize there are 18 ways the sum is even, and in only one of those is the blue die equal to 3 and the red die is 5, hence the probability is one out of eighteen.