A market maker sends 3 independent orders. They are filled with probabilities 1/2, 1/3 and 1/4 respectively. What is the probability that at least one order is filled?

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Chance of any fill

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A market maker sends $3$ independent orders. They are filled with probabilities $\frac{1}{2}$ , $\frac{1}{3}$ and $\frac{1}{4}$ respectively. What is the probability that at least one order is filled?

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Hint

Multiply the probabilities that each order is not filled.

Solution

The orders are not filled with probabilities $\frac{1}{2}$ , $\frac{2}{3}$ and $\frac{3}{4}$ .
By independence, none is filled with probability $\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} = \frac{1}{4}$ .
So at least one is filled with probability $1 - \frac{1}{4}$ .
This equals $\frac{3}{4}$ .

Answer: $\frac{3}{4}$

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