Expected Value for Decisions
Pantry Plus is a ₹199-a-month membership, and some members cancel every month. Marketing wants to send a ₹100 voucher to members who look likely to leave. Some of them would have stayed anyway, and some will leave no matter what. Is the voucher worth sending, and to whom? Expected value answers both questions with one line of arithmetic.
The rule
Value each outcome, weight it by its chance, and add them up.
The expected value of a choice is the average payoff you would get if you made that same choice many times:
where is the chance of outcome and is what it is worth. Pantry's numbers:
- A member who stays for another year is worth ₹1,000 in margin.
- The voucher changes the mind of 1 in 4 members who were about to cancel.
- Every member who receives it uses it, so it always costs ₹100.
For a member with a 60% chance of cancelling, the voucher saves them with probability 0.6 × 0.25 = 0.15. The expected gain is 0.15 × ₹1,000 = ₹150, minus the certain ₹100 cost: +₹50. Not sending changes nothing, so its expected value is ₹0. Sending wins.
For a member with only a 20% chance of cancelling: 0.2 × 0.25 × ₹1,000 = ₹50, minus ₹100, gives −₹50. Here the voucher mostly goes to people who were staying anyway.
The shortcut
Multiply along the branch that pays, then subtract what you pay for certain. Then find the break-even chance, where the two sides are equal. It turns one calculation into a rule for everyone.
The paying branch
Only one outcome gains money: the member would cancel and the voucher keeps them. Its value is risk × 0.25 × ₹1,000.
This rule is why Pantry needs a model that estimates each member's cancel risk. You will build that bridge in the model evaluation lessons.
The trap is to forget the members who would have stayed. They still redeem the voucher, and leaving their cost out makes every offer look better than it is. Remember too that expected value is an average over many members: any single member either stays or leaves.
Drills
Answer fast — trust the rule.
The expected value of sending a voucher to one group is +₹50 per member. What does that mean?
If the voucher cost fell from ₹100 to ₹50, the break-even cancel risk would:
A colleague counts the voucher cost only for members who would have cancelled. Their expected value will be:
Remember
- Weight each outcome by its chance, add them up, and compare the options.
- Multiply along the paying branch, subtract the certain cost, then solve for break-even.
- Trap: forgetting the people who would have behaved the same without you.