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### Course: Statistics and probability > Unit 9

Lesson 5: Binomial random variables- Binomial variables
- Recognizing binomial variables
- 10% Rule of assuming "independence" between trials
- Identifying binomial variables
- Binomial distribution
- Visualizing a binomial distribution
- Binomial probability example
- Generalizing k scores in n attempts
- Free throw binomial probability distribution
- Graphing basketball binomial distribution
- Binompdf and binomcdf functions
- Binomial probability (basic)
- Binomial probability formula
- Calculating binomial probability

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# 10% Rule of assuming "independence" between trials

10% Rule of assuming "independence" between trials.

## Want to join the conversation?

- doesn't this mean drawing cards in a deck of cards without replacing can be binomial if my number of trials is less than 5?(14 votes)
- I understand that it won't be binomial per definition, but can be treated as if it was because the results will be close enough to the precisely calculated numbers.(9 votes)

- How does the 10% rule make sense? The 10% rule says that if my sample size is less than 10% of the population, then I can assume independence. Isn’t this counterintuitive? Why does taking a smaller sample size result in a more accurate probability?(4 votes)
- Hi Rishabh,

A smaller sample size does not result in a more accurate probability, but rather results in the ability to assume independence, which then allows us to make some useful inferences about the results. Sal touches on this during the last minute.

Hope this helped. You can learn anything!

~Dayvyd(12 votes)

- why is 10% chosen? what advantage does it have over other percentages?(3 votes)
- The main idea here is that because as the proportion of the sample size over the population approaches 0, it behaves more like binomial distribution. So people might want to make a rule of thumb to use the assumption of independence. There's no particular reason to choose why 10% as why don't we choose 11% or 9%. It depends on the statistician's preference to accuracy. One possible reason to favor 10% is because it's easier to compute 10% of a number than, let say, 8%. Hope that helps! CMIIW(8 votes)

- what are the properties of normal distribution? there is no video till now on it , to best of my knowledge. Plz provide me link if there is so.(1 vote)
- Here are some properties of the normal distribution.

1. The normal distribution is symmetric about its only peak. The peak is located at the mean, median, and mode, which are all equal.

2. The probability is approximately 68% that the score is within 1 standard deviation from the mean (in either direction), approximately 95% that the score is within 2 standard deviations from the mean, and approximately 99.73% that the score is within 3 standard deviations of the mean.

3. Any linear combination of any number of independent normally distributed random variables is also normally distributed.

4. For a sufficiently large number of independent random variables with a common distribution (not necessarily normal) with finite mean and finite nonzero variance, the sample mean is approximately normally distributed.(10 votes)

- Does the 10% rule apply to a randomized experiment?(2 votes)
- Yes, the 10% rule can apply to randomized experiments as well, particularly when the experiment involves sampling from a larger population. The rule helps assess whether the sample size is sufficiently small relative to the population size to justify assuming approximate independence between trials.(1 vote)

- So does the 10% condition take into account future population? I'm analyzing data from all 29 kids in my senior class and we're a small school, so this is more than 10% of seniors who have gone to my school, but less than 10% of all seniors who will ever go to my school. Does that qualify or not?(2 votes)
- The 10% condition typically refers to the current population size from which the sample is drawn, rather than considering future populations. In your scenario, if you're analyzing data from all 29 kids in your senior class, and that represents more than 10% of the current seniors at your school but less than 10% of all seniors who will ever attend your school, it would still qualify for the 10% condition. The condition is based on the current population size to assess whether the sample size is reasonably small relative to that current population.(1 vote)

- In this example what was the sample size?(2 votes)
- The mall example would be 10,000, thee class example would be "n"(0 votes)

- In the example that Sal explains at the beginning about a mall, if there is people entering the mall while we sample the people that leave the mall, we shouldn´t mind about the 10% rule right? Because this way is as if there was replacement.(1 vote)
- Keep in mind that it's likely the person entering the mall would give a different answer than the person exiting. This could potentially make the difference between w/o and w/ replacement even greater than in the case where no one could enter.(1 vote)

- the video refers to the rule of independence but doesn't that rule just mean that the outcome of one trial doesn't affect the outcome of another?

i feel like the right rule this relates to is the one that states that the probability of success must be constant (which is usually what replacement affects)

maybe that was just an oversight but that was a bit confusing(1 vote)- You're correct that the rule of independence typically refers to the condition where the outcome of one trial does not affect the outcome of another. This is indeed related to the concept of independence in probability theory. The replacement aspect often affects the constancy of the probability of success on each trial, which is another important consideration for binomial distributions. While the video may have simplified by referring to the "rule of independence," it encompasses both the idea of trials being independent and the constancy of the probability of success on each trial.(1 vote)

- why is it important for our trials to be independent in a binomial varible?(1 vote)
- if the trials are independent, you can reasonably assume that the random variable is binomial(1 vote)

## Video transcript

- [Instructor] As we go further
in our statistical careers, it's going to be valuable to assume that certain distributors
are normal distributions or sometimes to assume that
they are binomial distributions because if we can do that, we can make all sorts of
interesting inferences about them when we make that assumption. But one of the key things
about normal distributions or binomial distributions is
we assume that they're the sum or they can be viewed as the sum of a bunch
of independent trials so we have to assume that trials are independent. Now that is reasonable
in a lot of situations, but sometimes let's say you're conducting a survey of people exiting a mall and in that case and
let's say you're saying whether they have done
their taxes already. If they're exiting the mall, it's hard to do these
samples with replacement. They're leaving the mall. You can't say, "Hey, hey, wait. "I just asked you a question. "Now you've answered it. "Now go back into the mall "because I want each trial
to be truly independent." But we all know it feels intuitive that hey if there are
10,000 people in the mall and I'm going to sample 10 of them, does it really matter that
it's truly independent? Doesn't it matter that we're just close to being independent? And because of that idea and because we do wanna make inferences based on things being close
to a binomial distribution or a normal distribution, we have something called the 10% rule and the 10% rule says that if our sample, if our sample is less than or
equal to 10% of the population then it is okay to assume
approximate independence and there are some fairly
sophisticated ways of coming up with this 10% threshold. People could have picked 9%. They could have picked 10.1%, but 10% is a nice round number. And if we look at some tangible examples, it seems to do a pretty good job. So for example right over here, let's let x be the number
of boys from three trials selecting from a classroom of n students where 50% of the class is a boy and 50% of the glass is a girl and so what we have over here is we have a bunch of different n's. What if we have 20 students in the class? What if we have 30? What if we have 100? What if we have 10,000? And so we could find the probability that we select three boys with replacement in each of these scenarios and we could also find the probability that we select three
boys without replacement and then we could think about what proportion is our sample
size of the entire population and then we could say, "Hey, does the 10% rule
actually make sense?" So this first column where we are picking three
boys with replacement, in this case because we are replacing, each of these trials are independent, are truly independent. And if our trials are independent, then x would be truly a binomial variable. Here, we aren't independent
because we are not replacing, so not independent, and so officially in this
column right over here when we're not replacing, x would not be considered
a binomial random variable. Let's see if there's a threshold where if our sample size is
a small enough percentage of our entire population
where we would feel not so bad about assuming x is
close to being binomial. So in all of the cases where
you have independent trials and 50% of the population
is boys, 50% is girls, you're going to amount to
1/2 times 1/2 times 1/2 so in all of those situations
we have a 12.5% chance that x is going to be equal to three and in this case x would
be a binomial variable. But look over here. When three is a fairly large
percentage of our population, in this case it is 15%, the percent chance of getting
three boys without replacement is 10.5% which is reasonably
different from 12.5%. It is 2% different but
2% relative to 12.5% so that's some place in
between 10 and 20% difference in terms of the probability. So this is a reasonably big difference. But as we increase the population size without increasing the sample size, we see that these numbers get closer and closer to each other all the way so that if you have 10,000 people in your population and you're only doing three trials that the numbers get very, very close. This is actually 12.49 something percent, but if you round to the
nearest tenth of a percent, you see that they are close. So I think most people would say, "All right, if your sample "is three ten-thousandths
of the population "that you'd feel pretty good "treating this column without replacement "as being pretty close to
being a binomial variable." And most people would say, "All right, this first scenario "where your sample size
is 15% of your population, "you wouldn't feel so good treating this "without replacement column as
a binomial random variable." But where do you draw the line? And as we alluded to earlier in the video, the line is typically drawn at 10%. That if your sample size is less than or equal to
10% of your population, it's not unreasonable to
treat your random variable, even though it's not
officially binomial to say, "Okay, maybe it is. "Maybe I can functionally
treat it as binomial "and then from there "I can make all of the
powerful interferences "that we tend to do in statistics." With that said, the lower the percentage the sample is of the
population the better. Now to be clear, that's not saying that small
sample sizes are better than large sample sizes. In statistics, large sample
sizes tend to be a lot better than small sample sizes. But if you wanna make this
independence assumption, so to speak, even when
it's not exactly true, you want your sample to be a small percentage
of the population. So the ideal, let's say you're
doing a survey at the mall, you might wanna survey 100 people but you would hope that there's at least 1,000 people in the mall
in order for you to feel like your trials are
reasonably independent. If there's 10,000 people in the mall or somehow 50,000 people in the mall, which would be a very large
mall, well that's even better.