Question . A coin box contains 8 fair, standard coins (head and tails) and 1 coin which has heads on both sides. A boy selects a coin randomly and flips it 4 times, getting all heads. If he flips this coin again, what is the probability it will be heads?
Let “Hi” be the event that Head occurs at the ith flip.. i=1(1)5
So the question asks for the conditional probability,
P(H5|H1, H2, H3, H4)
=P(H1, H2, H3, H4, H5)/P(H1, H2, H3, H4)
[The events “Hi’s” are independent]
[Where, “H” is the event of occurance of head]
=[P(H|fair coin selected)×P(fair coin selected)]+[P(H|unfair coin selected)×P(unfair coin selected)] [By theorem of Total Probability]
My problem is that for a coin that gives 4 heads as outcome, it is very probable that it is the biased coin and if I use this Classical definition, it is somehow not reflecting in the resultant probability of getting the 5th head..It is 5/9 which is near to 0.556.
But as getting 4 heads previously indicates high possibility of selectiin of biased coin, the 5th toss getting head should have a probability figure close to 1.
Can you state any other convenient way to solve this problem ?