Probability of getting tail with a coin toss using 2 coins A person has 2 coins one normal and one with HEAD on both sides. Now I dont know which coin he picked but the first THREE tosses are HEADs. Now whats the probability of seeing a tail next. 
I think since toss is independent event, at any point the probability is 1/4. 
Prob to pick good coin * prob of tail + prob of bad coin * prob of tail
1/2 * 1/2 + 1/2 * 0 = 1/4

Can anyone confirm if this is right or if there is a different solution. 
 A: You had no prior information about which coin it was, so suppose they were equally likely.
Three coin tosses have come up heads.  That would always happen with the second coin, but only happen once in eight with the first coin.  So the second coin is now eight times more likely than the first coin.
Put the new probabilities of the two coins into your formula.
A: Let the event "tails" be $T$, the event of choosing the good coin $G$, and three heads be $3H$ then 
$$
p(T|3H) = {1\over 2} p(G|3H) 
$$
and Bayes' theorem gives
$$
p(G|3H) = {p(3H|G) p(G)\over p(3H)} = {1/8\times1/2\over p(3H)}
$$
Now
$$
p(3H) = p(3H|G) p(G) + p(3H|{\rm not } G) p({\rm not } G)
$$
$$
= 1/8 \times 1/2 + 1\times 1/2
$$
The final answer is $1/18$.
A: If you want the easy solution and get an answer, use Baye's theorem and the answer is 1/18.
If you want to understand it, you need to split it. Answer those questions :


*

*What is the probability that he picked the first coin considering that he flipped 3 heads in a row?


To answer that first question you first need :


*What is the probability that he will flip 3 heads in a row if you don't know what coin he picked and both coin are as likely?


He has 50% chance to pick the first good coin (face and tail) and 50% change to pick the twisted coin (2 faces)
The probability to get 3 head in a row with the good coin is 1/8 and 1/1 with the twisted coin. So, the probability to get 3 heads in a row is 1/2*1/8 + 1/2*1/1 = 9/16
Then you need


*What is the probability to get 3 heads in a row AND pick the good coin?


1/2*1/8 = 1/16
Now... You got everything you need to answer the first question. The probability he picked the first coin considering he flipped 3 heads in a row is :
(#3) / (#2) = (1/16) / (9/16) = 1/9
So... Since he has 1/9 chance to have picked the first coin and has 1/2 chance to get tail on next turn if he picked the first coin, then he got
1/9 * 1/2 = 1/18 chance to get tail on next turn
Note : Sorry for the formating. I will figure how to use Tex ...
