Let’s first name our events so that we can use them easily.
Let A be the event coin 1 is selected
Let B be the event coin 2 is selected.
Let C be the event coin 3, that is the coin with both the faces head is selected.
Let H be the event that we get a head.
You pick a coin at random and toss it, and get heads. We want to know the probability that the coin picked up is C or the biased coin.
It can be represented as P(CH)
Using the formula for conditional probability, we get this as
P(CH)=P(H∩C)P(H)
P(H) is the probability of getting heads
We can get head in three different ways. We select A and get head or we select B and get head or we select C and get a head
⇒P(H)=P(HA)P(A)+P(HB)P(B)+P(HC)P(C)
=12×13+12×13+1×13
=23−−−−−−−−−−(1)
P(H∩C) can be calculated using the expression for finding
P(HC)P(HC)=P(H∩C)P(C)
⇒P(H∩C)=P(C)×P(HC)
P(HC) is the probability of getting head given that we chose coin C. Since coin C is biased, we get
P(HC)=1
⇒P(H∩C)=13×1
=13−−−−−−−−−−−(2)
Using (1)and (2)we get P(CH)=P(H∩C)P(H)
=1323
=12