Risk & Reward

Three or Two

Down by two with seconds left, you can take a 40 percent three to win outright or a 70 percent two to force overtime. Which shot gives your team the better chance?

solvedeasy1 min

Your team is down by two points with the ball and seconds left. A three-pointer wins the game outright and you sink it with probability 0.400.40. A two-pointer ties the game and sends it to overtime, which your team wins with probability 0.500.50, and you make the two with probability 0.700.70. Should you shoot the three or the two?

Reveal solutionHide solution

#Win probability of each shot

The three-pointer wins the instant it drops, so its win probability is just the make rate,

P(win∣three)=0.40.(1)\PP(\text{win} \mid \text{three}) = 0.40. \tag{1}

The two-pointer only ties, so you must first make it and then win the coin-flip overtime,

P(win∣two)=0.70⋅0.50=0.35.(2)\PP(\text{win} \mid \text{two}) = 0.70 \cdot 0.50 = 0.35. \tag{2}
threetwo then OT
The three-pointer wins straight away with probability 0.40. The two-pointer first has to drop and then survive a fifty-fifty overtime, leaving 0.35. The direct shot wins more often.

#Result

Since 0.40>0.350.40 > 0.35, take the three-pointer. The flashier-looking two has the higher make rate, but tying only buys a coin flip, which discounts that 0.700.70 down to 0.350.35 and below the direct three. Going for the win beats playing for the tie.