A 40% Shot Beats a 70% Shot That Only Buys a Coin Flip

Two points down, one second on the clock. The long shot goes in four times in ten and wins the game outright. The short shot goes in seven times in ten, but it only levels the score, and then the game is decided by an overtime you win half the time. Which do you take?

The long one, 0.40 against 0.35. The premise that pushes people the other way is completely true: seven in ten really is better than four in ten. What is wrong is the assumption that going in and winning are the same event.

One hurdle against two

The long shot has a single hurdle. Make it and the game is over:

The short shot has two, in sequence and independent of one another. It has to go in, and then the overtime has to go your way:

So the gap is $2/5 - 7/20 = 1/20$, five chances in a hundred, in favour of the harder shot. Exact, not a rounding: both numbers are fractions with small denominators and their difference is one twentieth.

How wrong the reflex is, in two currencies

A comparison of two numbers is less useful than knowing how much has to change before the verdict flips. Take the overtime win rate $q$ as unknown and solve:

The short shot needs an overtime you win 57.14 percent of the time to draw level with the long one. That is not a coin flip, and a team with no particular edge does not have it. The reflex is not narrowly wrong. It is wrong by about seven points of overtime strength, which is a large quantity in the only place it could come from.

Priced the other way, hold the overtime at a coin flip and ask how good the short shot would have to be, writing $m$ for its make rate:

Eighty percent from the field, against the seventy the problem grants. Both readings say the same thing and they say it in units the decision-maker actually has intuitions about.

The rule this is an instance of

In general, with $p_L$ for the long shot, $p_S$ for the short one and $q$ for the overtime, take the long shot when:

The left side is a ratio of shooting percentages and the right side is a property of the two teams, so the decision separates cleanly into something the shooter controls and something he does not. Here that ratio is $4/7$, and notice that it is the ratio and not the difference that carries the decision: adding the same ten points to both shooting percentages moves the comparison, while multiplying both by the same factor leaves it exactly where it was.

Written this way, the case for the long shot gets stronger the worse your team is in overtime, and weaker the better it is. A dominant team should take the tie. That is not a paradox: the overtime is a second chance whose value depends entirely on what you can do with it.

The objective that makes the reflex correct

The reflex is not stupid, it is optimising the wrong thing, and it is worth naming what it does optimise. Count expected points instead of expected wins:

The short shot scores more points on average, by a comfortable margin, and it still wins fewer games. Expected points is the right objective for most of a game, which is exactly why it is the habit a player brings to the last second, and at the buzzer with a two-point deficit it is the wrong one. Points beyond the ones you need are worth nothing, and points short of the ones you need are worth nothing either. The payoff function has stopped being linear, so the quantity that was worth maximising has stopped being the right quantity.

What is deliberately not in this model

The problem as posed has exactly two branches, and it is worth being explicit about the real basketball it leaves out. A missed long shot cannot be rebounded and put back, because there is one second left. A made short shot cannot win, because the deficit is exactly two. And the overtime is handed to us as a number rather than modelled, which hides the fact that a team with the ball trailing by two is usually not the stronger team.

The branch that would matter most in practice is the foul. A long attempt that draws a foul converts into free throws with a very different distribution, and shooting fouls on long attempts are not rare. Nothing above rules that out; it is simply outside the two-branch problem, and adding it means adding a probability the question never supplied. The honest version of the answer is conditional: given these three numbers and no other branches, 0.40 beats 0.35.

One thing the model does not need is any assumption about correlation between the shot and the overtime, which would otherwise be a fair objection to (2). It does need them independent, and that is stated. If making the short shot somehow raised the overtime win rate, the product in (2) would rise, and (3) tells you exactly how far it would have to rise to matter.

Sources and further reading

• Chaining the two stages of the short shot — Chain rule (probability)
• Splitting a decision by what happens next — Law of total probability
• The quantity being compared in (5) — Expected value
• Drawing the branches instead of arguing about them — Decision tree

Every figure here is an exact fraction. On top of that, four hundred thousand simulated possessions played out both decisions including the overtime as two separate draws, so the 0.35 was produced by playing the game rather than by multiplying two numbers: the long shot measured 0.39946 and the short shot 0.35059. A ten-thousand-point sweep of the overtime rate located the crossover at 0.5715 against the exact $4/7 = 0.571428$ without being told where to look.