Probability 6

Mixing multiple probabilities

Given a box of 10 coins out of which 9 coins are fair and one coin has both sides heads, what is the probability of getting all heads on picking a coin from the bag and flipping it 5 times.

There are two outcomes possible if we pick a coin from the bag.
The first outcome would be that we pick a normal coin and its probability would be 9/10 --- (1).
The second outcome would be that we pick a coin having both sided heads, whose probability is 1/10 --- (2).

Now after this,
Given a normal coin what is the probability of getting all 5 heads is, as we know it to be (1/2)*(1/2)*(1/2)*(1/2)*(1/2) = 1/32 --- (3).

Given a both sided heads coin, the probability of getting all 5 heads is 1 --- (4).

From (1), (2), (3) & (4) we get the
P(getting all 5 heads) = (P(picking the normal coin)*P(getting all heads provided its normal coin))+(P(Picking an unfair coin)*P(getting all heads provided its unfair coin))

= (1/32)*(9/10) + 1*(1/10) = 9/320 + 1/10 = 41/320