Chance a trader's 64-trade total dips low

Question

A trader's individual trades average a profit of $\$ 5 $with a standard deviation of$ $40$ each. Trades are independent, so by the central limit theorem the total profit over many trades is approximately normal.

Over $n = 64$ trades, what is the probability the total profit is below $-\$ 160$?

Accepted Answer

We want the total across 64 trades, so mean and variance both add. The total profit is approximately normal with 	ext{mean} = n\mu = 64 	imes 5 = \320, 	ext{SD} = \sigma\sqrt{n} = 40 	imes \sqrt{64} = 40 	imes 8 = \320. Find the z-score of the -\160 threshold: z = \frac{-160 - 320}{320} = \frac{-480}{320} = -1.5. From the normal table: P(	ext{total} < -160) = P(Z < -1.5) \approx 0.0668, about 6.7\%. Despite a positive expected profit of \320, the large per-trade swings leave a real chance of ending well underwater over 64 trades. Takeaway. For a sum, mean = n\mu and SD = \sigma\sqrt{n}:…

Chance a trader's 64-trade total dips low

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