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Naveen's coin box contains 8 fair standard coins (heads and tails) and 1 coin which has heads on both sides. He 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?

My solution: Coin tosses are independent.
P(Head) = P(Head/Fair Coin)P(Fair Coin) + P(Head/Unfair Coin)P(Unfair Coin)
= 1/2 * 8/9 + 1 * 1/9
= 4/9 + 1/9
=5/9

Am I right?

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You have not made use of the fact that the coin he picked came up heads 4 times. You need to introduce an extra term in each summand of your computation for the chance that this kind of coin will come up heads four times in a row.

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Guide.

Let $H_i$ denote the event that the $i$-th throw results in a head.

Let $E$ denote the event that a fair coin was selected.

To be found is $P(H_5\mid \bigcap_{i=1}^4H_i)$ which can be done on base of:

  • $P(H_5\mid \bigcap_{i=1}^4H_i)=P(\bigcap_{i=1}^5H_i)/P(\bigcap_{i=1}^4H_i)$

  • $P(\bigcap_{i=1}^nH_i)=P(\bigcap_{i=1}^nH_i\mid E)P(E)+(\bigcap_{i=1}^nH_i\mid E^{\complement})P(E^{\complement})$ for every $n$.

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