Traders A and B each wait for a fill. In every round, independently of everything else, A's order fills with probability 1/3 and B's order fills with probability 1/3. Let X and Y be the round numbers of their first fills. What is P(X = Y)?

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Simultaneous first fills

ProbabilityHard

Traders A and B each wait for a fill. In every round, independently of everything else, A's order fills with probability $\frac{1}{3}$ and B's order fills with probability $\frac{1}{3}$ . Let $X$ and $Y$ be the round numbers of their first fills. What is $P (X = Y)$ ?

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Hint

Sum over the common round number $n$ the probability that both first fills happen in round $n$ .

Solution

For one trader, $P (X = n) = (\frac{1}{3}) \times (\frac{2}{3})^{n - 1}$ .
By independence, $P (X = n and Y = n) = (\frac{1}{9}) \times (\frac{4}{9})^{n - 1}$ .
Summing over $n = 1, 2, 3, \dots$ gives a geometric series with first term $\frac{1}{9}$ and ratio $\frac{4}{9}$ .
Its sum is $(\frac{1}{9}) \div (1 - \frac{4}{9}) = (\frac{1}{9}) \div (\frac{5}{9}) = \frac{1}{5}$ .
So $P (X = Y) = \frac{1}{5}$ .

Answer: $\frac{1}{5}$

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