Probabilistic approach to calculating expected value

Question

Why is it correct to use these terms in the following calculation of upside/downside/expected value?

Suppose the chance of missing a meeting is 20% and there is a 25% chance the client will sign a big deal at this meeting. The deal itself is worth $1 million.

upside value = (1 - 0.2) * 0.25 * 1000000 = $200,000

if we miss the meeting there's a 95% chance the client will break relations

downside value = (1 - 0.2) * 0.95 * 0.25 * 1000000 = $47500

expected value = 200000 - 47500 = $152500

$152500 in favor of trying to make the meeting.

What would be an example that favor's skipping the meeting?

Answer 1

I am not sure what you mean by upside and downside in your expected value calculation, but the probability of the deal going through and the expected value of the event, $E$ can be calculated as follows. From the law of total probability

$$P(D) = P(D|M)P(M) + P(D|\neg M)P(\neg M)$$

where $P(D)$ is the probability of making the deal and $P(M)$ is the probability of missing the meeting. The values can be calculated from the given information, so $$ P(D) = 0.05 \times 0.2 + 0.25 \times 0.8 = 0.21.$$

Now, there are 4 possible outcomes for the event $E$. The expected value is the outcome times the probability of each outcome where the outcome is gaining 1 million or gaining 0, so \begin{align} E[E] &= 1 \times P(D \cap M) + 1 \times P(D \cap \neg M) + 0 \times P(\neg D \cap M) + 0 \times P(\neg D \cap \neg M) \\ E[E] &= P(M)P(D|M) + P(D)P(\neg M| D) \\ E[E] &= 0.2 \times 0.05 + 0.21 \times (1-0.0476) = 0.21 \end{align}

where $P(\neg M| D) = 1 - P(M|D)$ and $P(M|D)$ can be calculated from Bayes' rule. So, the expected value is 0.21 million, so \$210,000.

Answer 2

The upside of making the meeting is: the oportunity to sign the deal : $$\mathsf E(X^+)=80\%\cdot25\%\cdot\$1\,000\,000$$

The downside of missing the meeting is: the possibility of a breakup if you miss the meeting, the cost of which shall be evaluated as the loss of oportunity to sign the deal: $$\mathsf E(X^-)=20\%\cdot 95\%\cdot25\%\cdot\$1\,000\,000$$

So the expected opportinity for scheduling the meeting is: $$\mathsf E(X)=\mathsf E(X^+)-\mathsf E(X^-)$$

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An example of favour of skipping a meeting is one where there is a downside to making the meeting and an upside to missing it. Say, the possibility of the contract might actually wind up costing much more than its worth.