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Quote fills in a session

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A market maker posts a quote in $5$ independent rounds, and in each round it is filled with probability $0.2$ . What is the probability that it is filled in at least $2$ rounds?

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Hint

Use the complement: subtract the probabilities of $0$ fills and of exactly $1$ fill.

Solution

The probability of $0$ fills is $0. 8^{5} = 0.32768$ .
The probability of exactly $1$ fill is $5 \times 0.2 \times 0. 8^{4} = 0.4096$ .
The complement gives $1 - 0.32768 - 0.4096 = 0.26272$ .
The probability is $0.26272$ .

Answer: $0.26272$

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