# Statistics: Experimental Probability.

#### killivolt

Joined Jan 10, 2010
835
My Daughter wanted to know if she was correct, she has a class online and the professor is making the students work in groups to answer this question. She wanted to know if, 0.14 is correct.

Thanks,
kv

#### SamR

Joined Mar 19, 2019
4,811
14/100 = 0.14 = 14% looks right to me...

#### WBahn

Joined Mar 31, 2012
29,156
If I were grading it and they say that it is 0.14 I would find that highly suspicious, actually. This would be like asking someone to flip a coin 100 times and tell me how many times it ended up heads. If they come back and say that they got 50 heads and 50 tails, I would suspect that they didn't actually do the experiment, but rather just calculated the expected result and claimed that that was what they got (based on the 90+% likelihood that they would NOT have gotten 50 heads and 50 tails).

What she has done is calculate the THEORETICAL PROBABILITY, but that is not what she was asked to do. She is very specifically tasked with finding the EXPERIMENTAL PROBABILITY. The point of experimental probability is that it allows us to estimate the theoretical probability of systems that are either too complex or that we don't have sufficient details about in order to calculate the theoretical probability.

She she needs to perform some experiment that will allow her to estimate the probability based on the results of the experiment.

The easiest and most compelling would be to get some Skittles and set up a bag as described and then do a bunch of trials recording how many times she gets a purple Skittle. Since she is part of a team and since each team member can perform trials independently, they should be able to collect a fair amount of data pretty quickly. If they plot their estimate as a function of the number of trials performed, then they should see very erratic results that eventually start to settle down. This can give them a feel for how many trials are needed before they can start to have confidence that their experimental probability is becoming a reasonable estimate.

What I would do if I were in their place is do both a physical experiment and collect as much data as I reasonably could (and I would record the actual colors and the order in which they were drawn) and then do a computer simulation of the process and overlay the physical data on the simulation data.

#### jpanhalt

Joined Jan 18, 2008
11,087
The observed result was 14%. I suspect the professor wants the students to calculate/estimate a theoretical confidence range for that result. For example, one could make an assumption that there were equal numbers of each color in the container from which the skittles were taken. What is the probability one would see that distribution in that event. What if one asked only, what is the probability of drawing 18 reds in 100 grabs without regard to the other colors?

Some rules of thumb may help see that. One is that the standard deviation for any sample is roughly the square root of that sample (e.g, the S.D. for red is √18 ≈ 4.2). For quick estimates without calculations , I round up. Thus the expected range for red is 18 ± 2xS.D. That is, if you repeated the experiment several times, the number of reds would be expected to be between (18 - 2xS.D.) and (18 + 2xS.D) 95% of the time. (Assuming everything is fair.)

I notice the reference book is a nursing manual and suspect the professor's intent is along those lines. How deeply into the mathematics have they gone?

#### killivolt

Joined Jan 10, 2010
835
LOL, I'm sure her children would love to do that experiment

But, these are Nursing students taking a Statistics class, I made sure she writes the equation correctly (P) = 1/5 chances. I'm not sure where she got the bag holds 100 skittles, so they used that as the amount of pulls from the bags to equal the amount of occurrences given. It would be cool to do a graph or other representations as a functional result of different students with separate results. I was surprised with the actual occurrences equal 100 if you total them. So, I qustioned it would be impossible to get the resultant amount of occurrences with only 100 draws.

kv

#### killivolt

Joined Jan 10, 2010
835
The observed result was 14%. I suspect the professor wants the students to calculate/estimate a theoretical confidence range for that result. For example, one could make an assumption that there were equal numbers of each color in the container from which the skittles were taken. What is the probability one would see that distribution in that event. What if one asked only, what is the probability of drawing 18 reds in 100 grabs without regard to the other colors?

Some rules of thumb may help see that. One is that the standard deviation for any sample is roughly the square root of that sample (e.g, the S.D. for red is √18 ≈ 4.2). For quick estimates without calculations , I round up. Thus the expected range for red is 18 ± 2xS.D. That is, if you repeated the experiment several times, the number of reds would be expected to be between (18 - 2xS.D.) and (18 + 2xS.D) 95% of the time. (Assuming everything is fair.)

I notice the reference book is a nursing manual and suspect the professor's intent is along those lines. How deeply into the mathematics have they gone?
You beat me LOL

kv

#### jpanhalt

Joined Jan 18, 2008
11,087
It took me more than an hour to write that drivel. I write more slowly than most people can imagine, but it is a subject medical personnel need to have a gut feeling for. In fact, I did virtually that same experiment with coffee beans for med tech students.

#### WBahn

Joined Mar 31, 2012
29,156
Ah, I misread the problem. I thought it was giving the actual distribution and then asking them to devise an experiment and take data. But I see that it is saying that the provided information is the data that was presumably collected from some prior experiment.

So, given that understanding, then the experimental probability if, indeed, 14%.

#### WBahn

Joined Mar 31, 2012
29,156
LOL, I'm sure her children would love to do that experiment

But, these are Nursing students taking a Statistics class, I made sure she writes the equation correctly (P) = 1/5 chances. I'm not sure where she got the bag holds 100 skittles, so they used that as the amount of pulls from the bags to equal the amount of occurrences given. It would be cool to do a graph or other representations as a functional result of different students with separate results. I was surprised with the actual occurrences equal 100 if you total them. So, I qustioned it would be impossible to get the resultant amount of occurrences with only 100 draws.

kv
I don't see anything that indicates that the bag has 100 Skittles in it. It appears that it is a bag that contains SOME Skittles and that some reached in, picked on out, recorded the color, and returned it to the bag. They then repeated this 100 times and the data given is just the total observed results. All we know for sure about the bag is that it has at least five Skittles in it (since five different colors were observed over the course of the trials).

#### jpanhalt

Joined Jan 18, 2008
11,087
From what I have read (Google), Wrigley's claims that an equal number of each color/flavor is made. However, there are several ".edu" studies that indicate yellow (lemon) may predominate in the individual packages. If an average commercial package (by weight) is about 50 to 70 skittles, this problem resolves to putting confidence limits on the predicted composition of the source of, say, 100 packages. This problem is a bit odd, in that 100 skittles are sampled, so I assume the author is assuming an equal distribution of colors/flavor for instructional purposes..

I am not an affectionaire of skittles and have never bought them. Do they come commercially in larger packages, say of 1000 or more?

#### WBahn

Joined Mar 31, 2012
29,156
This problem is a bit odd, in that 100 skittles are sampled, so I assume the author is assuming an equal distribution of colors/flavor for instructional purposes..
I'm still not seeing where it is implied that the population consists of 100 Skittles. Since it is supposedly data from experimental observations, it seems far more likely that we have the tabulated results of 100 experiments, probably performed with replacement.

#### jpanhalt

Joined Jan 18, 2008
11,087
"Experiment" is singular. Experimental could mean the results of many experiments.