How do I figure out the odds of...

NickThieves

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May 12, 2008
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3
First a disclaimer:

Hi, I'm NickThieves, undereducated and quite possibly posting in the wrong section. I don't speak mathematical technobabble, so if you, dear reader, attempt to help me out, please try to speak English or all you'll achieve is to make me look ignorant, which I freely admit that I am so you needn't bother ;)

Now to my problem:

Say you have 12 targets and 12 shots.

Each shot has 94% chance of hitting a random target.

Assuming I know what I'm doing, that means the 2nd shot has ca. 8% chance of hitting the same target as the 1st shot.

Unfortunately, I don't know how to figure out the chances of how subsequent shots will be distributed, so that's what I'd like your help with. Pretty please?
 
NickThieves said:
First a disclaimer:

Hi, I'm NickThieves, undereducated and quite possibly posting in the wrong section. I don't speak mathematical technobabble, so if you, dear reader, attempt to help me out, please try to speak English or all you'll achieve is to make me look ignorant, which I freely admit that I am so you needn't bother ;)

Now to my problem:

Say you have 12 targets and 12 shots.

Each shot has 94% chance of hitting a random target.

Assuming I know what I'm doing, that means the 2nd shot has ca. 8% chance of hitting the same target as the 1st shot.

Unfortunately, I don't know how to figure out the chances of how subsequent shots will be distributed, so that's what I'd like your help with. Pretty please?

Unfortunately, I do not see a question anywhere (other than that "pretty please?").
 
Subhotosh Khan said:
Unfortunately, I do not see a question anywhere (other than that "pretty please?").
[
tkhunny said:
Why are successive shots NOT independent?
I'll try to explain myself a little better then.

12 shots are fired simultaneously at 12 targets.
The target of each shot is randomly selected.
Each shot has 94% chance to hit a target.

How do I calculate the distribution of the hits on the targets?

Obviously the chances of all 12 targets being hit are tiny, but how do I calculate how tiny that chance is? How do I calculate the chance of a specified target amongst the 12 getting hit more than once, more than twice, more than thrice and so on?

I'm not asking for a particular result, I'm asking how to calculate the probabilities of all the possible ways the hits can be distributed on the targets.
 
Hello, NickThieves!

12 shots are fired simultaneously at 12 targets.
The target of each shot is randomly selected.
Each shot has 94% chance to hit a target.

How do I calculate the distribution of the hits on the targets?

(a) Obviously the chances of all 12 targets being hit are tiny,
. . but how do I calculate how tiny that chance is?
. .
I think I have this one . . .

(b) How do I calculate the chance of a specified target getting hit
. . more than once, more than twice, and so on?
. .
This is an awesomely difficult problem . . .

\(\displaystyle \text{(a) First, all 12 targets must be chosen.}\)

. . \(\displaystyle \begin{array}{cc}\text{The first shot can be at any of the 12 targets: } &\frac{12}{12} \\ \\[-3mm] \text{The 2nd shot must be at one of the other 11 targets: } & \frac{11}{12} \\ \\[-3mm] \text{The 3rd shot must be at one of the other 10 targets: } & \frac{10}{12} \\ \vdots & \vdots \end{array}\)

\(\displaystyle P(\text{all 12 targets chosen}) \;=\;\frac{12}{12}\cdot\frac{11}{12}\cdot\frac{10}{12}\cdots \frac{3}{12}\cdot\frac{2}{12}\cdot\frac{1}{12} \;=\;\frac{12!}{12^{12}} \;=\;\frac{11!}{12^{11}}\)

. . \(\displaystyle \text{Then each shot must hit its intended target: }\:(0.94)^{12}\)


\(\displaystyle \text{Therefore: }\;P(\text{all 12 targets hit}) \;=\;\frac{11!}{12^{11}}(0.94)^{12} \;\approx\;0.000025568\)

 
Thanks a lot Soroban. Oh, and thanks for introducing me to Factorials and weird uses of exclamation marks. Never knew about those, haha.

I also think you got part (b) of my question right, even though I phrased it hopelessly. To clarify, each round of 12 shots will be fired any number of times, so what I'm asking for is a way to calculate the frequency of the various ways each round of shots can hit & miss the 12 targets, so that I can determine how likely it is that a target will be hit X, Y or Z times in a particular round of shots.

...Man, I really do hope I managed to clarify it.
 
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