[Math] Conditional probability that the first toss resulted in heads

probability

A fair coin is tossed until two heads have appeared.

  1. Given that exactly $k$ tosses were required, what is the conditional probability that the first toss resulted in heads?
  2. If $p_k$ is the probability that at least $k$ tosses are required, find a formula for $p_k$ and find the smallest $k$ such that $p_k\le0.1$.

How do I approach problems like this? For the first question I am not able to apply the Bayes/Price theorem because I am not sure how to derive the $P(A\cap B)$ expression in the numerator. For the second, I am stuck at "at least $k$ tosses".

Best Answer

  1. Given that exactly $k$ tosses were required, what is the conditional probability that the first toss resulted in heads?

Yes, you use Bayes' formula: Let $H_n$ count the trials until head $\#n$ occurs.

$$\mathsf P(H_1=1\mid H_2=k) ~=~ \dfrac{\mathsf P(H_2=k\mid H_1=1)~\mathsf P(H_1=1)}{\mathsf P(H_2=2)}$$

Can you now find these probabilities from first principles?

  1. If $p_k$ is the probability that at least $k$ tosses are required, find a formula for $p_k$ and find the smallest $k$ such that $p_k \leqslant 0.1$.

$$p_k~=~\mathsf P(H_2\geq k) ~=~ 1-\mathsf P(H_2<k)$$

That is: $p_k$ is the probability that you get no more than one head among the first $k-1$ tosses.

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