Can someone explain the thought process used when solving th

Can someone explain the thought process used when solving this question. Thank you!

I have two dice in my pocket. One die has four sides labeled with the numbers 1,2,3,4, the other has six sides labeled 1,2,3,4,5,6. In both cases if I roll a die all numbers are equally likely to come up I choose a die at random (both dice being equally likely), and roll it. What is the probability I chose the four sided die, given that I rolled a 1?

Solution

Let

A = chose first die
B = chose second die
1 = rolled a 1

Thus,

P(A|1) = P(A) P(1|A) / P(1)

However, by Bayes\' rule,

P(1) = P(A) P(1|A) + P(B) P(1|B) = (1/2)*(1/4) + (1/2)*(1/6) = 0.208333333

Thus,

P(A|1) = P(A) P(1|A) / P(1)

= (1/2)*(1/4)/0.208333333333

= 0.6 [ANSWER]