# Thread: Calculating overall percentage/probability from multiple categories?

1. ## Calculating overall percentage/probability from multiple categories?

Greetings,

My apologies if the title is confusing; I don't really know how to explain what I am trying to do or what label this problem would fall under.

Scenario:
We have a magic bag, and inside the magic bag are an unknown/unlimited amount of coins.
There are 100 different types of coins, but we are only interested in the iron, bronze, silver and gold coins, so we have bundled the other 96 types together as "other".

We performed an experiment by taking one coin at a time from the magic bag and recording what type of coin it was; the coins we took from the magic bag were not placed back into the magic bag.
We performed the experiment 1,000,000 times and the results are as follows:

Other coins: 998,859
Iron coins: 596
Bronze coins: 312
Silver coins: 135
Gold coins: 98

First Question:
Is it correct to say the probability to receive each coin from the magic bag is as follows?

Other coins: (998,859/1,000,000)*100 = 99.8859% or 1,000,000/998,859 = 1 in 1.0011423
Iron coins: (596/1,000,000)*100 = 0.0596% or 1,000,000/596 = 1 in 1,678
Bronze coins: (312/1,000,000)*100 = 0.0312% or 1,000,000/312 = 1 in 3,205
Silver coins: (135/1,000,000)*100 = 0.0135% or 1,000,000/135 = 1 in 7,407
Gold coins: (98/1,000,000)*100 = 0.0098% or 1,000,000/98 = 1 in 10,204

Second Question:
If I take only one coin from the magic bag, what is the chance/probability to receive either an iron, bronze, silver or gold coin?
(Receiving any of these four coins would be a success, and receiving any of the other 96 coins would be a failure).

I have tried to do some calculations, but I don't think I am working it out properly.
-Is the following correct?
((0.0596/100)+(0.0312/100)+(0.0135/100)+(0.0098/100))*100= 0.1141% (or 100% - 99.8859% = 0.1141%)

-Is the following correct?
((1/1678)+(1/3205)+(1/7407)+(1/10204))*100= 0.1141%

-Is the following correct?
(((1/1678)+(1/3205)+(1/7407)+(1/10204))/4)*100= 0.0285%

Third Question:
If I take 1,400 coins from the magic bag, what is the chance/probability to receive either an iron, bronze, silver or gold coin?
(Receiving any of these four coins would be a success, and receiving any of the other 96 coins would be a failure).

I really have no idea how to calculate this; all I have managed to do is repeat one of the formulas above and multiply by 1,400.

-Is the following correct?
((((1/1678)+(1/3205)+(1/7407)+(1/10204))/4)*100)*1400= 39.9339%

Fourth Question:
Whatis the name for this type of probability?

I understand that I am probably completely wrong about everything, so thank you very much to anyone willing to provide assistance.

2. ## Re: Calculating overall percentage/probability from multiple categories?

Answer to First Question: No. You only have one experiment. The bag is magic. Perhaps among the first million coins, it was magically preprogrammed to produce those specific coins with each pull. Perhaps the next million coins are all gold. It seems unlikely, but it is possible. The best you can say is that you likely found the probabilities for each pull. Other statistics could give you how confident you should be with the probabilities you generated. But that is a slightly more complicated topic that I will not get into in this post.

Answer to Second Question: You are correct the first two ways you try it. The third way (dividing by 4) is incorrect. You are not looking for the average probability, you are looking for the total probability. Addition is correct when dealing with disjoint outcomes and looking for the probability that any of them could happen. If you had mixed metal coins (like half-gold, half-silver, and it is counted as both gold and silver), then the probabilities would be a bit more difficult. But taking an average is not correct.

Answer to Third Question: I am not sure what you mean. You are taking 1,400 coins. There is a zero percent chance that after taking 1,400 coins, you will have a single coin of any type. You will have 1,400 coins. If you mean that among the 1,400 coins, the probability that at least one coin will be of the appropriate type, then the answer is:

$$1-\left(\dfrac{998,859}{1,000,000} \right)^{1,400} \approx 0.797762395008639443151242821398077659595272041337 934988070 = 79.77623950086394431512428213980776595952720413379 34988070\%$$

It is 1 minus the probability that every coin is in the "other" category.

Answer to Fourth Question: Binomial Probability or Bernoulli Trials: https://en.wikipedia.org/wiki/Binomial_distribution