Strategy, Risk, and the Analyst's Stance

Lesson 2 of 2: Two-Sided Strategy and the Limits of Expected Value

In this lesson:

• Compare a strategy by its net effect on winning
• Weigh risk and asymmetric error costs
• Communicate a recommendation responsibly

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Compare a strategy by its net effect on the probability of winning
2. Recognize when risk or asymmetric error costs override expected value
3. State a recommendation with its assumptions and limitations

A Third Lens: Net Effect on Winning

Testing used conditional probability. Now a new lens:

Compare strategies by their net effect on the probability of winning.

Down a Goal, Late: Keep or Pull?

Your team trails by one goal with under a minute left.

Keep the goalie in, or pull them for an extra attacker?

Goalie In: A Low Chance of Tying

With the goalie in:

• You attack normally — but you're a man short on offense
• Your chance of tying in the time left is low

Goalie Out: Both Probabilities Move

The extra attacker raises your tying chance — and raises the opponent's empty-net chance.

Compare the Net Win-or-Tie Chance

For each strategy, find the probability of at least tying (keeping a chance to win).

Recommend the strategy with the higher win-or-tie probability.

The Trap of One-Sided Analysis

Counting only your extra scoring chance overstates the gamble

A strategy that helps you often helps the opponent too — model both sides

It Depends on Score and Time

The right call depends on the situation:

• Down by one with a minute left — pulling is often right
• Down by three, or five minutes left — the math changes

Your Turn: Which Strategy Wins?

Goalie in: tying chance .
Goalie out: tying chance , but empty-net loss chance .

Which gives the higher win-or-tie probability?

When Expected Value Isn't the Criterion

So far: average it, or read a conditional probability.

But sometimes the average is the wrong thing to optimize entirely.

Risk: Equal Means, Different Worst Case

Two strategies can share an average yet differ sharply in their worst outcome.

Error Costs Are Often Asymmetric

In cancer screening, a false negative (missed case) is far costlier than a false positive.

Some Payoffs Are Not Monetary

Health, safety, and reputation are not dollars.

Sometimes the "payoff" in the frame is a value judgment, not a number.

Pulling the Whole Domain Together

You can now:

• Define random variables and compute expected values
• Build distributions, score and compare strategies
• Design fair procedures and analyze decisions with risk

Your Turn: When Does Risk Override?

A clinic chooses a screening threshold. Lowering it catches more cancers but raises false alarms.

Why might lowering the threshold be right even if total errors rise?

The Stance of a Responsible Analyst

A responsible recommendation always:

• States its assumptions
• Shows its probabilities
• Names what it does not capture

What You Learned This Lesson

✓ Compare strategies by their net effect — model both sides

✓ Let risk and asymmetric error costs override the average when they should

✓ Recommend with assumptions, probabilities, and limits stated

Where These Ideas Go Next

These ideas grow into decision theory — Bayes' theorem, ROC curves, utility.

And into real policy: screening rules, recalls, risk regulation.