Average number of different colors of marbles in a sample?

[JS-Student]

Hi, I'm asking this in relation to samples of DNA amplicons, but I'm so confused by probability, that I think it would be easier for me to understand in terms of simple marbles-in-bag situation.

Say there is a bag of 10 different colors of marbles in equal amounts, and you are allowed to pull out 12 marbles, each time with replacement. In your sample of 12 marbles, how many different colors of marbles would you expect to have?

If someone could walk me through this, especially the concepts necessary to figure this out, I would greatly appreciate it.

 

[micromass]

This page contains the answer: http://www.math.uah.edu/stat/urn/Birthday.html

 

[JS-Student]

Thank you, I will try to understand this tutorial.

 

[micromass]

Thank you, I will try to understand this tutorial.

Please do ask if something is unclear. The result you want is ##E(V)## in ##19##. The values are ##m=10## and ##n=12##.

 

[blue_raver22]

Hello,

Thank you for your question. I can provide a response to your query regarding the average number of different colors of marbles in a sample.

In your example, we have a bag with 10 different colors of marbles in equal amounts. This means that each color has an equal chance of being picked when you draw a marble from the bag. This concept is known as probability, where we calculate the likelihood of an event occurring.

Now, let's consider the situation where you are allowed to pull out 12 marbles, each time with replacement. This means that after each draw, the marble is returned to the bag, and the same 10 colors are still present in the bag. In this case, the probability of picking a specific color remains the same for each draw.

To calculate the expected number of different colors in a sample of 12 marbles, we can use the concept of expected value. This is the average number of different colors we would expect to see in a sample of 12 marbles, if we were to repeat this experiment many times.

The formula for expected value is: E(X) = n * p, where n is the number of trials and p is the probability of an event occurring. In this case, n = 12 (number of draws) and p = 1/10 (probability of picking a specific color). Therefore, E(X) = 12 * 1/10 = 1.2.

This means that on average, we would expect to see 1.2 different colors in a sample of 12 marbles. Since we cannot have a fraction of a color, we can round this number to the nearest whole number, which is 1. Therefore, we would expect to see 1 different color in a sample of 12 marbles.

I hope this explanation helps to clarify the concept of probability and expected value for you. If you have any further questions, please feel free to ask. Thank you.