Question

There are three coins in a box. One is a two-headed coin, another is a fair coin, andthe third is a biased coin that comes up heads 75 percent of the time. When oneof the three coins is selected at random and ipped, it shows heads. What is theprobability that it was the two-headed coin?

Answer 1

Answer:  The probability that it was the two-headed coin is $\dfrac{4}{9}$ .

Step-by-step explanation:

Let we consider the events as

A =  A two-headed coin is tossed

B =  A fair coin is tossed

C = biased coin is tossed

H= Results heads after toss

As per given , total coins =3

So , $P(A)=P(B)=P(C)=\dfrac{1}{3}$

Probability that coins show heads :

$P(H|A)=1\\P(H|B)=\dfrac{1}{2}\\P(H|C)=0.75$

By Bayes theorem ,we have

$P(A|H)=\dfrac{P(H|A)}{P(A)P(A|H)+P(B)P(B|H)+P(C)P(C|H)}\\\\=\dfrac{1(\dfrac{1}{3})}{1(\dfrac{1}{3})+\dfrac{1}{3}(\dfrac{1}{2})+\dfrac{1}{3}(0.75)}$

Simplify , we get    $\dfrac{4}{9}$

Hence, the probability that it was the two-headed coin is $\dfrac{4}{9}$ .