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Question

A box contains three coins: two regular coins and one fake two-headed coin (P(H)=1)
You pick a coin at random and toss it, and get heads. The probability that it is the two-headed coin = ___

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Solution

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(HC)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(HC) can be calculated using the expression for finding
P(HC)P(HC)=P(HC)P(C)
P(HC)=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(HC)=13×1
=13(2)
Using (1)and (2)we get P(CH)=P(HC)P(H)
=1323
=12

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