Target this, part 2

Probability Level 3

After extensive time at the archery range, Tom, Dick and Harry have all improved their shooting. Now Tom has a 20% chance of hitting the bulls-eye with every shot, and Dick and Harry are even better than that.

They continue to play the same game, with them all putting money in the can and the first to hit the bulls-eye winning the money. They continue to use the same order: Tom first, then Dick, then Harry, and cycle in that order repeatedly until someone hits the bulls-eye.

All three of them now have the same chance of winning the money. What is Harry's chance to hit the bulls-eye in percent, to 2 decimal places?

The answer is 33.33.

2 solutions

Denton Young
Jun 12, 2017

Consider 500 cycles. Tom goes first, and wins 100 of them. Dick has to win 100 of the remaining 400 cycles, leaving 300 cycles in which we get to Harry. And Harry has to win 100 of those. So Harry's shooting accuracy must be $100/300 = 1/3 = 33.33$ %

Uros Stojkovic
Jun 23, 2017

Since all three of them have the same chance of winning the money, we can set up an equation:

$t=(1-t)d=(1-t)(D-1)h$ ,

where $t$ , $d$ and $h$ are chances of Tom, Dick and Harry in decimal form. Plugging in $t=0.2$ , we get:

$0.2=(1-0.2)d=(1-0.2)(D-1)h$

Solving for $d$ , we get $d=\frac{0.2}{0.8}=0.25$ . Now, we have:

$h=\frac{0.2}{0.8\times 0.25}=0.333...$ - expressed in percent, to two decimal places: $h=33.33$ %.

Target this, part 2

The answer is 33.33.

2 solutions

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