Sale or hang-up first?

Question

A salesperson works a call that passes through states. From the Start of a call, the customer either agrees to buy (a Sale, probability $0.3$ ) or moves to a hesitant state $A$ (probability $0.7$ ). From $A$ the customer either hangs up (probability $0.5$ ) or returns to the Start of the pitch (probability $0.5$ ). Sale and hang-up end the call.

Starting from Start, what is the probability of a Sale before a hang-up?

Accepted Answer

Let f(X) be the probability of a Sale before a hang-up from state X, with f(	ext{Sale}) = 1 and f(	ext{hang-up}) = 0. Use first-step analysis: each state's value is the average of where the next step goes. f(A) = 0.5 \cdot 0 + 0.5\,f(	ext{Start}) = 0.5\,f(	ext{Start}), f(	ext{Start}) = 0.3 \cdot 1 + 0.7\,f(A) = 0.3 + 0.7\,f(A). Substitute the first into the second: f(	ext{Start}) = 0.3 + 0.7\,(0.5\,f(	ext{Start})) = 0.3 + 0.35\,f(	ext{Start}). So 0.65\,f(	ext{Start}) = 0.3, giving f(	ext{Start}) = \frac{0.3}{0.65} = \frac{6}{13} \approx 0.4615. So a Sale comes before a hang-up with …

Sale or hang-up first?

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