How do I figure out the odds of...

[NickThieves]

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?

 

[Deleted member 4993]

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?").

 

[tkhunny]

Why are successive shots NOT independent?

 

[NickThieves]

Unfortunately, I do not see a question anywhere (other than that "pretty please?").

[

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.

 

[soroban]

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 . . .

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

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

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

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


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

 

[NickThieves]

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.