Probability Calculation

In summary, the probability of getting 12 different colors out of 16 when picking 16 marbles at random from an infinite mix of marbles with 16 possible colors is approximately 0.1303. This can be calculated by finding the number of ways to order 12 distinct colors out of 16 and dividing it by the total number of possible sequences of 16 colors.
  • #1
davee123
672
4
Suppose I have an infinite mix of marbles, each of which can be one of 16 colors. You pick 16 marbles at random. How do you calculate the probability of receiving, say, 12 different colors within the 16 you picked?

The probability of getting 1 color is pretty easy: 16/16^16

And the probability of getting all 16 colors is also easy: 16!/16^16

But the probability of getting something in the middle, I'm fuzzy on how to calculate. I wrote a script to test, so I know the rough probability of getting 12 colors is 13.04%, but I can't quite wrap my brain around how to calculate the more accurate probability, if (say) the number of picks changed, or if the number of colors changed.

DaveE
 
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  • #2
davee123 said:
Suppose I have an infinite mix of marbles, each of which can be one of 16 colors. You pick 16 marbles at random. How do you calculate the probability of receiving, say, 12 different colors within the 16 you picked?

The probability of getting 1 color is pretty easy: 16/16^16

And the probability of getting all 16 colors is also easy: 16!/16^16

But the probability of getting something in the middle, I'm fuzzy on how to calculate. I wrote a script to test, so I know the rough probability of getting 12 colors is 13.04%, but I can't quite wrap my brain around how to calculate the more accurate probability, if (say) the number of picks changed, or if the number of colors changed.

DaveE
Hi DaveE,

I get a probability of 0.13029987, which is pretty close to your value of 0.1304.

You can possibly draw any of 16^16 sequences of colors, each of which we assume is equally likely.

There are [tex]\binom{16}{12}[/tex] ways to select 12 colors out of the 16.

Let's say there are n ways of ordering 16 colors, 12 of which are distinct; then the probability we seek is
[tex]p = \frac{\binom{16}{12} n}{16^{16}}[/tex]

It remains only to find n. I used an exponential generating function for this, but there are sure to be other ways. Let's say [tex]a_r[/tex] is the number of ways to order a selection of r colors drawn from a pallet of 12. Let
[tex]f(x) = \sum_{r=0}^{\infty} \frac{1}{r!} a_r x^r[/tex]
Then (it's easy when you know how!) we see
[tex]f(x) = (x + (1/2) x^2 + (1/3!) x^3 + \dots)^{12}[/tex]
[tex]= (e^x -1)^{12}[/tex]
[tex]= \sum_{i=0}^{12} (-1)^i \binom{12}{i} e^{ix}[/tex]

From this we can see that
[tex]n = a_{16} = -\binom{12}{1} + \binom{12}{2} 2^{16} - \binom{12}{3} 3^{16} + \dots + \binom{12}{12} 12^{16}[/tex]
which is approximately 1.3206639 x 10^15. Substituting this value for n in the equation for p above, we find p is approximately 0.13029987, as claimed.
 

FAQ: Probability Calculation

What is probability calculation?

Probability calculation is the process of quantifying the likelihood of a certain event or outcome occurring. It involves using mathematical techniques to analyze data and make predictions about the probability of certain events happening.

How is probability calculated?

The probability of an event occurring is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This can be represented as a fraction, decimal, or percentage.

What are the different types of probability?

There are three main types of probability: theoretical, experimental, and subjective. Theoretical probability is based on mathematical principles and is calculated by analyzing the possible outcomes of an event. Experimental probability is based on actual data collected through experiments or observations. Subjective probability is based on personal beliefs or opinions about the likelihood of an event occurring.

What are some common applications of probability calculation?

Probability calculation is used in a wide range of fields, including statistics, finance, economics, and science. It is used to make predictions, assess risk, and make decisions based on data.

What are some common misconceptions about probability calculation?

One common misconception is that probability can predict the outcome of a single event. In reality, probability can only provide an estimate of the likelihood of an event occurring based on available data. Another misconception is that past events can influence the probability of future events. In reality, each event is independent and the probability remains the same regardless of past outcomes.

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