# How did Cowen and Tabarrok calculate the probability of Hewlett-Packard sales?

The discussion on prediction markets in the chapter on the price system in Tyler Cowen and Alex Tabarrok's Modern Principles of Economics, Second Edition (pp. 123-124) contains the following paragraphs:

Members of HP’s sales team bought and sold shares that paidoff when sales fell within a certain range. A typical security would pay out $$\\\1$$, if and only if future sales were, say, between $$10,000$$ and $$15,000$$ units. Another might pay off if sales were between $$15,000$$ and $$20,000$$ units. The market con- tained $$10$$ types of securities—a range broad enough to include all the relevant possible sales outcomes.
By examining the prices of all $$10$$ shares, HP could assign a probability to any combination of outcomes. For example, if the price of the $$5,000–10,000$$ unit sales security was $$10$$ cents and the price of the $$10,000–15,000$$ unit sales security was $$20$$ cents, this suggests that the probability of selling $$5,000 –15,000$$ units was $$30\%$$.

How did the authors arrive at this figure of $$30\%$$?

• If people are willing to pay only 10 cents for a chance of getting a dollar, that means they think they're about 10% likely to get a dollar. Commented Apr 12, 2021 at 9:24

It is this probability that is consistent with shares trading for their expected value. The expected value is given by: $$E(S_1+S_2)=E(S_1)+E(S_2)=p_1 \times1+p_2 \times1=p_1+p_2$$

where $$S_i$$ is the payment of share $$i$$, $$p_i$$ is the probability of payment of share $$i$$ (due to the sales figure falling in some range). So if the expected value is to equal the price of the security, then:

$$E(S_1+S_2)=P_1+P_2 \Rightarrow p_1+p_2=P_1+P_2$$

Where $$P_i$$ denotes the price of share $$i$$. So the probability of one of them paying off equals the sum of their prices (note that the events of share 1 and 2 paying off are mutually exclusive).

$$P_1=\\\0.20$$
$$P_2=\\\0.10$$