Find probability of biased Coins

Question

There are two biased coins A and B. We have been given the following:

P(H|Coin A) = 0.9
P(T|Coin A) = 0.1
P(H|Coin B) = 0.1
P(T|Coin B) = 0.9

We are also given that Probability of tossing Coin A or Coin B randomly is 0.5

How could we find the probability of "11th toss = Head", given that we have 10 heads in a row ?

This depends on whether you select the coin just once, or anew on each trial. — Graham Kemp, Jun 23 '17 at 08:56
We are selecting the coin just once. I would love to see how answer changes if we select the coin anew on each trial — user3036345, Jun 23 '17 at 09:34

score 3 · Accepted Answer · answered Jun 23 '17 at 06:40

Given we have coin A, the probability of tossing 10 Heads in a row is $0.9^{10}$; given we have coin B, the probability of tossing 10 Heads in a row is $0.1^{10}$. Using Bayes rule we have

$$P(A|10H)=\frac{P(A)P(10H|A)}{P(10H)}=\frac{0.5\cdot 0.9^{10}}{0.5\cdot 0.1^{10}+0.5\cdot 0.9^{10}}=\frac{0.9^{10}}{0.1^{10}+0.9^{10}}$$

$$P(B|10H)=\frac{P(B)P(10H|B)}{P(10H)}=\frac{0.5\cdot 0.1^{10}}{0.5\cdot 0.1^{10}+0.5\cdot 0.9^{10}}=\frac{0.1^{10}}{0.1^{10}+0.9^{10}}$$

The probability $P$ that the 11th toss is Heads given 10 previous tosses were Heads is

$$P=P(A|10H)\cdot 0.9 + P(B|10H)\cdot 0.1$$

wonderful - Thanks for your explanation – user3036345 Jun 23 '17 at 06:45 — user3036345, Jun 23 '17 at 06:45

Find probability of biased Coins

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