# Probability for 20 people

#### ae5

##### New Member
Hi,

I'm not a stats student, I'm a university student reviewing some literature and I'm interested in some of the statistics related to it and how they've been reinterpreted and published elsewhere. I'd really appreciate your help on this.

I used to study stats but it's been a few years so I'm not so sure of it.

I'm looking at something where there is a 5% chance of 1 person getting a particular disease. I can't remember the term but it happening to one person doesn't affect the probability of it happening to someone else, so each are independent of each other.

I read an interpretation of this statistic which is that 1 in 20 people will get the disease. To my understanding that's wrong, because there's a 5% chance for each person and that doesn't mean 1 person out of 20 will get it. To me, if there are 20 people, it's a 5% chance for each. So, I decided to try to work this out statistically to confirm it. This is what I got.

Probability 0 out of 20 people get the diease = (0.95)^20 = 0.358 = 35% probability 0 out of a group of 20 get the disease

Probability 1 out of 20 people get the diease = 0.05 x (0.95)^19 = 0.05 x 0.377 = 0.0189 = 2% 1 in a group of 20 people get the disease

Is this correct or am I working this out wrong?

Values are to 3 significant figures except percentages which I put in integer values.

If this isn't correct, please let me know what I'm doing wrong . If it is, then the above work out at about 37%. Where would the other 43% of possibilities come from? I haven't tried working out for 2 - 20 people getting the disease but given that there's such a low probability of one getting the disease I imagine these will be low percentages.

Thank you

#### Dason

When you specify the probability there will be 1 person with the disease you need to multiply by 20. There are (20 choose 1) possible ways that 1 person out of the 2 could get the disease.

#### ae5

##### New Member
Hi, thanks for that. I remember nCr, which is I think what you mean?

If that's the case, I'm trying to imagine using it and I seem to remember that, long hand it would be something like:

0.05x(0.95)^19 + 0.95x0.05x(0.95)^18 + (0.95)^2x0.05x(0.95)^17 ... + (0.95)^19x0.05

Is that right? If so, given that each of those equals approx. 2%, wouldn't the probability be 2% x 20 = 40%?

#### squareandrare

##### New Member
I read an interpretation of this statistic which is that 1 in 20 people will get the disease. To my understanding that's wrong, because there's a 5% chance for each person and that doesn't mean 1 person out of 20 will get it.
What they should have said is that 1 in 20 people on average will get the disease.

What you're describing is the Binomial distribution.

The probability that exactly X out of the 20 people have the disease is:
$$\frac{20!}{(20-X)!X!}(.05)^X(.95)^{20-X}$$

#### ae5

##### New Member
Hi, thanks for that. The equation is great. I assumed (hopefully correctly) that that's calculating the probability that X people get the disease out of 20.

So I then used it for 1 out of 20 and got 0.377 / 37.7% and for 0 out of 20 and got 0.358 / 35.8%.

So, from that, that means the probability of, out of 20 people, 1 getting the disease is ~ 38%?

Does that make the common practice of taking something like 5% chance and translating it to 1 in 20 people, or even an average of 1 in 20 people, a bit misleading?

Just, average 1 in 20 sort of sounds like it's very, very likely that 1 in 20 people will get the disease, but from the formula, the actual chance of any 20 people having 1 get the disease is only 38%. This is without other factors of course.

My reasons for this is I'm discussing how facts in research or statistics/numbers can be distorted in the media, to give an inaccurate impression to the public. To me, 38% chance of 1 in 20 getting a disease sounds a lot less likely (as is an individual 5% chance) than "1 in 20". The article didn't say an average of 1 in 20 like you say squareandrare, but even if it did, I still think 'on average 1 in 20' suggests it's a lot more likely than it is statistically, if 38% is correct.

Thanks.

#### squareandrare

##### New Member
When they say "1 in 20", they don't mean for an actual sample of size 20. They're saying that if many people are sampled, you'll find about 1 case of the disease for every 19 you don't find.

I don't find the phrase misleading. It's a statement about the population proportion, not the proportion of a small sample.