Problem
Given 5 coins:
- 2 double-headed coins
- 2 fair coins
- 1 double-tailed coin
A coin is chosen at random and flipped. The coin is then flipped in a second time. What's the probability that face showing is a head?
My attempt was using conditional probability on each type of coin:
P( 1st = whatever, 2nd = head | DH ) * P( DH )
+ P( 1st = whatever, 2nd = head | DT ) * P( DT )
+ P( 1st = whatever, 2nd = head | FC ) * P( FC )
Apparently, P( 1st = whatever, 2nd = head | DT ) = 0. So I ended up with: $1 \cdot \dfrac{2}{5} + \dfrac{1}{2} \cdot \dfrac{2}{5} = \dfrac{3}{5} $
But my teacher said the answer is $\dfrac{5}{6}$ which I couldn't understand how did he come up with that solution! So my question is, are the 1st flip and 2nd flip two independent events? I don't see how should we take the 1st flip into account?