The Good, the Bad,
and the Ugly
In a re-creation of a famous 3-cornered duel (which occurs near the end of the movie, The Good, the Bad, and the Ugly) three friends are engaged in a paintball duel.
When they fire their paintballs, the probabilities that they will hit their target are:
The Ugly, 30%; the Bad, 50%; and the Good, 100% (the Good never misses!).
The Ugly will take the first shot, next the Bad will shoot, and finally the Good. Players will continue to take one shot at a time, continuing in this same order until there is only one player left unpainted, who will then be declared the winner. Players will always try to
take a shot that gives them the best chance of survival.
What should the Ugly’s first shot be?
Some Thoughts:
Let’s put ourselves in Ugly’s position.
Suppose that Ugly takes the first shot at Bad.
If Ugly hits Bad, as Ugly has a 30% chance of doing, then Bad is out. But then Good will hit Ugly with the next shot in the round, since Good has 100% accuracy!, and Ugly will lose.
If Ugly misses Bad, then Good will eliminate Bad, and Ugly will have one more shot at Good, with a 30% chance to win.
Suppose that Ugly takes the first shot at Good.
If Ugly misses Good, then Good will eliminate Bad, and Ugly will have one more shot at Good, with a 30% chance to win.
If Ugly actually hits Good, then a most interesting scenario could unfold. Bad will have the next shot, with a 50% chance of ending the game by hitting Ugly. But if Bad misses, then Ugly will get another shot, with a 30% chance of hitting Bad. But if Ugly then misses, Bad
will get yet another shot at Ugly, with a 50% chance of success. But if Bad misses—and so on. So this contest could go on and on!
Let’s list all the ways that Ugly could win. Using the notation BM for “Bad missed Ugly,” UM for “Ugly missed Bad,” and UHfor “Ugly hits Bad,” Ugly wins if the string of paintball shots is any of these:
and so forth…. There are infinite outcomes for this branch of the game. The question is, if Ugly gets into this duel, can Ugly beat his 30% survival rate by missing his first shot entirely? Stay tuned….
More Ponderings:
Recall that Ugly has the first shot, then Good gets a shot, and Bad is up last. Probabilities of a hit are 0.3 for Ugly, 1.0 for Good, and 0.5 for Bad. In last month’s discussion of this Problem to Ponder, we reasoned that if Ugly missed the first shot (either at Bad or at Good), then he had a
30% chance to win, because Good would hit Bad with his paint ball for certain on his turn, and Ugly would then have a 30% chance to hit Good on the next turn. On the other hand, if Ugly actually hit Good on his first shot, the resulting duel between Bad and Ugly could go on for some time. We need to look at all the
ways that Ugly could win in the scenario of the Bad vs. Ugly duel.
Ugly can win the duel with any sequence of paintball shots that starts with a miss by Bad and ends with a hit by Ugly. Using the notation BM for “Bad missed Ugly,” UM for “Ugly missed Bad,” and UHfor “Ugly hits Bad,” Ugly wins with any of these sequences of paintball shots:
BM UH
BM UM BM UH
BM UM BM UM BMUH
BM UM BM UM BMUM BM UH
. . . .
An infinite number of such sequences end with Ugly hitting Bad after both miss a number of times. The probabilities that Bad and Ugly miss on any one paintball attempt are, respectively, P(BM) = .5, and P(UM) = .7.
The sequences that end with Ugly winning have the following probabilities:
P(BM UH ) = .5 x .3 = .15
P(BM UM BM UH)= .5 x .7 x .5 x .3 = (.35) x.15 = .0525
P(BM UM BM UM BMUH) = .5 x .7 x .5 x .7 x .5 x .3 = (.35)2 x .15 = .018375
P(BM UM BM UM BMUM BM UH) = .5 x .7 x .5 x .7 x .5 x .7 x .5 x .3 = (.35)3 x .15 = .00643125
. . . and so on.
Each of these sequences represents a different way that Ugly could win the contest, and adding the probabilities yields .15 + .0525 + .018375 + .00643125 = .22730625.
If we continue to add up allthe probabilities of all the (infinite number of) ways that Ugly could win, will it be greater than 30%? Adding all the probabilities for Ugly to win yields the following infinite series:
Σ [(.35)nx (.15)] , for n = 0, 1, 2, ….
Factoring out the constant term gives us
(.15) x Σ (.35)n .
The probability that Ugly wins in the Bad vs. Ugly duel leads to the geometric series
Σ (.35)n with ratio (.35), and therefore sums to 1/(1 – .35) = 1/(.65).
Substituting gives us
(.15) x 1/(.65). = (.15/.65) = 3/13.
Thus, the probability that Ugly wins in the Bad vs. Ugly duel scenario is 3/13, which is less than the 30% (or 3/10) probability that Ugly wins if he just misses in the first place.
So, what should Ugly do with his first paintball shot? Aim away from Good and Bad!