When does the sure score win?
Compare two guaranteed points with the expected payoff from two checks.
The question
A fictional game gives you two choices. Use the expected score to choose between them.
- Guaranteed
- Take 2 points.
- Two checks
- Each passes with probability 1/2. Each pass earns 4 points.
- Deduction
- If both fail, subtract 6 points once.
- Opening model
- The checks are independent.
“Which choice gives the higher expected score?”
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A strong answer, step by step
Open each step in turn. This is one strong answer, not the only one.
Step 1: Which result incurs the deduction?
What would you do here? Open the move you’d make.
Deduct six for every failed check
Tempting, but it misses something. The rules give no deduction for a single failure.
Give zero points when both fail
Tempting, but it misses something. That drops the six-point deduction.
Deduct six only when both checks fail
The strong move. This applies the deduction once to the stated outcome.
See how a strong candidate says it
“Only the outcome where both checks fail loses six points. One pass earns four, and two passes earn eight.”
Why it works: The deduction depends on the pair of results. A single failure carries no separate charge.
Step 2: Which choice has the higher mean?
What would you do here? Open the move you’d make.
Run the checks for 2.5 expected points
The strong move. That is higher than the guaranteed two points.
Run the checks for 4 expected points
Tempting, but it misses something. Four is the average before the joint-failure deduction.
Take two because either check can fail
Tempting, but it misses something. The question asks you to maximize the mean, which already includes failures.
See how a strong candidate says it
“The two checks earn 4 points on average before the deduction. Both fail with probability 1/4, so the expected deduction is 1.5 points. The net is 2.5 points. I’d take the checks.”
Why it works: The two individual gains contribute four points. Independent failures give a one-in-four deduction.
Step 3: Which choice wins now?
A shared outcome makes both checks pass or both fail together. Each still passes with probability 1/2.
What would you do here? Open the move you’d make.
Run the checks. They still average 2.5.
Tempting, but it misses something. The old calculation treats the two failures as independent.
Take two points. The checks now average one.
The strong move. A half chance of losing six lowers the mean to one.
Take two points. The checks earn no gains now.
Tempting, but it misses something. The new rule still permits both checks to pass and earn eight.
See how a strong candidate says it
“Both checks now fail together half the time. The gains still average four points and the deduction averages three. The net is 1 point. I’d take the guaranteed two.”
Why it works: The same individual pass chances leave the average gains unchanged. The shared result changes the chance of the deduction.
Step 4: Which part of the mean changed?
What would you do here? Open the move you’d make.
Multiply the two failure probabilities again
Tempting, but it misses something. The revised results are tied to the same outcome.
Treat any dependence as a reason to take two
Tempting, but it misses something. The stated deduction and its probability determine this choice.
Keep the gains and recompute the joint deduction
The strong move. This identifies the term that changed the decision.
See how a strong candidate says it
“The expected gains add to four under either rule. Only the joint deduction changes. Under the new rule the two payouts are eight and minus six, each half the time.”
Why it works: Linearity gives the same expected gains. It does not make the joint failure probability independent.
The answer, in short
- Decision
- Choose the checks under independence and the guaranteed points under the shared outcome.
- Because
- The joint deduction increases enough to reverse the mean payoff comparison.
- What would change it
- With a different deduction or guarantee, compare the means again.
Common traps
- Leaving the joint deduction out of the mean
- Multiplying marginal failure chances after the checks become dependent
Check your answer
- How does each pass contribute?
- When does the deduction apply?
- Which probability changes?
- Which option has the higher revised mean?
Last reviewed Nov 11, 2026