16 replies to this topic

[EventHorizon]

Spoiler for I looked at this puzzle last night...

and I came up with the same probabilities as curr3nt.

I thought perhaps combining the two darts would still have a larger variance and that would change things. I worked up formulas for the binomial distribution, combined them, and ended up with another binomial distribution of the new probability .98% (which really should have been obvious from the Bernoulli trial).

There surely was something I was missing.

Since then I've decided that....

Spoiler for Alex is a MACHINE!

I noticed the word reproducibly, and that changes things since if he never gets less than 97% we are no longer dealing with the binomial distribution for him.

Let B(k;n,p) be the probability of getting k successes in a Binomial distribution with n trials and a probability p of success.

Assuming he always gets exactly 97 out of 100, he will still lose since the probability the duo get 98, 99, or 100 is (B(100;100,.98) + B(99;100,.98) + B(98;100,.98)) which is slightly over 67%.



But will Alex ever get more than 97 out of 100?
Assuming Alex's distribution is uniform (so he gets 97, 98, 99, and 100 darts out of 100 with equal probability)...

The probability the duo will win is:
.25 * ( B(100;100,.98) + B(99;100,.98) + B(98;100,.98) ) +
.25 * ( B(100;100,.98) + B(99;100,.98) ) +
.25 * ( B(100;100,.98) ) +
.25 * 0
= about .303144

The probability they tie is :
.25* ( B(100;100,.98) + B(99;100,.98) + B(98;100,.98) + B(97;100,.98) )
= about .214740

This leaves the probability Alex win at .482116.
But that only works if Alex's distribution is uniform between 97 and 100 when throwing 100 darts.



But what if they only throw 33 darts at a time? Alex must always get all 33 or he will not have reproduced the result of a percentage greater than or equal to 97%. In this case he will never lose.

So that's my guess. They always throw darts in groups of 33. Alex always gets 97% or more, so he cannot miss. Alex gets some more beer money.

Though perhaps I'm reading too much into it...

[ujjagrawal]

When Davey and Ian playing together:

Combined probability of hitting a bull's eye will be -

90% + 10% x 80% = 98% (If Davey fails to hit, when going first)
or
80% + 20% x 90% = 98% (If Ian fails to hit, when going first)

Clearly, more than 97% probability by Alex, so Alex definitely made a BAD bet.

[mrzinks]

Spoiler for Hmmm.....

Wouldn't Davey and Ian's average accuracy be 95%, or 2% less than Alex's ??

[Rockmond]

Curr3nt did a lot of figuring to come up with the same conclusion I got by multiplying .1 by .2 and comparing it to .03. On 100 throws Alex loses a net 1 time.

[Rockmond]

When Davey and Ian playing together:

Combined probability of hitting a bull's eye will be -

90% + 10% x 80% = 98% (If Davey fails to hit, when going first)
or
80% + 20% x 90% = 98% (If Ian fails to hit, when going first)

Clearly, more than 97% probability by Alex, so Alex definitely made a BAD bet.

Easier to just calculate odds of neither Davey or Ian hitting bullseye: .1 * .2 = .02 or 2%

[jim]

Suppose they each throw 100 times with 97 bullseyes for Alex while Davey gets 90 and Ian gets 80. What would the team of Davey and Ian score? Suppose Davey throws first and misses 10 times. Ian lets the pressure get to him and misses every time Davye does.lus 10 additional times. In this case the team scores 90. On the other hand, if Davey is good under pressure and makes all 10 of those throws while missing on 20 other occassions, then the team scores 100. If the correlation coefficent for Davey and Ian is zero, then the team scores 98 and they win. With a negative correlation they do even better, but with a large enoiugh positive correlation the bet will be a good onwe for Alex.

[TheChad08]

Suppose they each throw 100 times with 97 bullseyes for Alex while Davey gets 90 and Ian gets 80. What would the team of Davey and Ian score? Suppose Davey throws first and misses 10 times. Ian lets the pressure get to him and misses every time Davye does.lus 10 additional times. In this case the team scores 90. On the other hand, if Davey is good under pressure and makes all 10 of those throws while missing on 20 other occassions, then the team scores 100. If the correlation coefficent for Davey and Ian is zero, then the team scores 98 and they win. With a negative correlation they do even better, but with a large enoiugh positive correlation the bet will be a good onwe for Alex.

You're using factors that aren't given to us and over examining the question.

What is 1 + 1?
Answer is 2
But what if I add to the question (like you) and turn it into 1 half + 1 half.
Then the answer would be 1.