A signal triggers N trades, where N is equally likely to be 1, 2 or 3. Each trade wins with chance 1/3, independently of the others. What is the chance that no trade wins? Give a fraction in lowest terms.

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A signal triggers $N$ trades, where $N$ is equally likely to be $1$ , $2$ or $3$ . Each trade wins with chance $\frac{1}{3}$ , independently of the others. What is the chance that no trade wins? Give a fraction in lowest terms.

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Hint

Condition on $N$ : given the count, all the trades must lose.

Solution

Given $N$ , every trade must lose, with chances $\frac{2}{3}$ , $\frac{4}{9}$ and $\frac{8}{27}$ for the possible counts.
These add up to $\frac{2}{3} + \frac{4}{9} + \frac{8}{27} = \frac{38}{27}$ .
Each count has weight $\frac{1}{3}$ , so the answer is $\frac{38}{27} \times \frac{1}{3} = \frac{38}{81}$ .

Answer: $\frac{38}{81}$

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