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

Question

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 $30\%$?

Answer 1

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).

In your case

$P_1=$$\\\$$$0.20$

$P_2=$$\\\$$$0.10$

Of course these calculations totally disregard the many complications that go into asset pricing (like discounting cash flows, accounting for having to wait for payment), but I assume the authors want to keep it simple to prove a point. The bigger picture you should take away from this answer is that share prices can be interpreted as reflecting information about (what investors believe to be) probabilities of future events.