Probability of flipped coin A bag has 3 coins a, b and c with probability of heads 0.9, 0.6 and 0.5.
I flip a randomly drawn coin and get heads.
What is the probability that it will come up heads again once re-flipped ?
probability of coin being 'a' is $\frac{0.9}{0.9+0.6+0.5}$ = 0.45

probability of heads in next turn = 0.9

.

probability of coin being 'b' is $\frac{0.6}{0.9+0.6+0.5}$ = 0.3

probability of heads in next turn = 0.6

.

probability of coin being 'c' is $\frac{0.5}{0.9+0.6+0.5}$ = 0.25

probability of heads in next turn = 0.5

Total probability = (0.45*0.9)+(0.3*0.6)+(0.25*0.5) = 0.71
\begin{align}
Ans = \bbox[yellow,5px,border:2px solid red]{0.71}
\end{align}
\begin{align}
P(\text{Next will be head}|\text{Current is head}) 
&amp;= \frac{P(\text{Next will be head} \cap \text{Current is head}) }{P(\text{Current is head}) } 
\\&amp; = \frac{P(\text{Next will be head})P(\text{Current is head})}{P(\text{Current is head})}
\\&amp;= P(\text{Next will be head}) 
\\&amp;= P(a)P(a|a\text{ is head}) + P(b)P(b|b\text{ is head}) + P(c)P(c|c\text{ is head})
\\&amp;= \bbox[yellow,5px,border:2px solid red]{\frac{1}{3}0.9 + \frac{1}{3}0.6 + \frac{1}{3}0.5}
\end{align}