Sales calls to close three deals

Question

A telemarketer closes each cold call independently with probability $1/10$ . The daily quota is $3$ closed sales, and calling continues until the quota is met.

On average, how many calls does it take to reach the third sale?

Accepted Answer

Reaching 3 sales is three independent single-sale waits chained together, each with success probability p = 1/10 and expected length 1/p. Adding them by linearity of expectation, E = \frac{r}{p} = \frac{3}{1/10} = 3 \cdot 10 = 30 	ext{ calls.} Takeaway. Expected calls to the r-th sale is r/p = 3 \cdot 10 = 30.

Sales calls to close three deals

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