# Thread: Binomial and conditional prob.

1. ## Binomial and conditional prob.

Q. What is the chance that in 10 flips of a coin, no heads will come up? What is the conditional probability that, after 10 straight heads,
the 11th flip will be a tail (e.g, Prob(11th flip is a tail | 10 heads in a row)?

My answer to first question is 10C0 x 0.5^0 x 0.5^10 = 0.0009765625.
However, My stuck question is the 2nd, the conditional probability. How can I approch to the 2nd question? Please tell me how to answer to the 2nd.

p.s. Prob(11th flip is a tail | 10 heads in a row) = Prob(11th flip is a tail AND 10 heads in a row / Prob(10 heads in a row).
Prob(10 heads in a row) = 0.0009765725, as answered above. right? and then, what is Prob(11th flip is a tail AND 10 heads in a row?

2. Originally Posted by ilovepsycho
Q. What is the chance that in 10 flips of a coin, no heads will come up? What is the conditional probability that, after 10 straight heads,
the 11th flip will be a tail (e.g, Prob(11th flip is a tail | 10 heads in a row)?

My answer to first question is 10C0 x 0.5^0 x 0.5^10 = 0.0009765625.
However, My stuck question is the 2nd, the conditional probability. How can I approch to the 2nd question? Please tell me how to answer to the 2nd.

p.s. Prob(11th flip is a tail | 10 heads in a row) = Prob(11th flip is a tail AND 10 heads in a row / Prob(10 heads in a row).
Prob(10 heads in a row) = 0.0009765725, as answered above. right? and then, what is Prob(11th flip is a tail AND 10 heads in a row?
Yes, your first question is correct.

"Prob(11th flip is a tail AND 10 heads in a row)" basically means that... suppose H= Heads and T=Tails. Then the event you are looking for is HHHHHHHHHHT.

You can use the geometric distribution formula: P(X=k) = p*(1-p)^k where p is the probability of success and there are k failures before your first success. So P(X=10)=0.5*(1-0.5)^10=0.5^11 is your answer to Prob(11th flip is a tail AND 10 heads in a row). So divide the answer by the answer you obtained in the first question to get your final answer.

Even without doing any computation, intuitively one can see the answer is 50% because the question is basically asking "Given that you got 10 heads in a row, what is the probability your 11th trial will result in a tail." Since coin flipping is independent, it doesn't matter what your previous results were.