Answer:
The probability that we have a fair coin is 0.2717
Step-by-step explanation:
Let Ci be the event that the coin i is used for while we let i = 1,2,3 (representing the three coin)
We also let H be the event that the coin which is flipped lands (which is head)
Therefore, the problem arise:
That: P(H/C1) = 1/2.............fair coin
p(H/C2) = 2/3..........Chance of head
p(H/C3) = 2/3............Chance of tail
Now, noting that the coin was picked at random,
We have, : p(Ci) = P(C1) = P(C2) = P(C3) = 1/3
We can then say that or calculate thus:
P( C2 | H ) = P (H ∩ C2) ÷ P(H)
Where P (H ∩ C2) means the probability of event intersection
= P(H ∩ C2) ÷ P(H ∩ C1) + P(H ∩ C2) + P(H ∩ C3)
= P(H | C2) P(C2) ÷ P(H | C1) P(C1) + P(H | C2) P(C2) + P(H | C3) P(C3)
= (1 / 2) (1 / 3) ÷ (1/2)(1/3) + (2/3)(1/3) + (2/3) (1/3)
0.5 × 0.3 ÷ (0.5 × 0.3) + (0.67 × 0.3) + (0.67 × 0.3)
0.15 ÷ 0.15 + 0.201 + 0.201
=0.15 / 0.552
= 0.2717
