I've got confused while understanding the concept of independent events when cases are randomly selected: like the problems below.

A poll finds that 72% of Jacksonville consider themselves football fans. If you randomly pick two people from the population, what is the probability the first person is a football fan and the second is as well?

Solution: one person being a football fan doesn’t have an effect on the second because they are randomly selected. Therefore, the events are independent and the probability can be found by multiplying the probabilities together: First one and second are football fans: P(A∩B) = P(A) · P(B) = .72 * .72 = .5184.

*my question is how the first person doesn't have effect on the second? if the first person is a football fan that means we left one football fan from the population. *


1 Answer 1


You made a great observation there and you are correct in general. Fortunately the effect is incredibly small for football fans in Jacksonville. Let's walk through it to see:

The population of Jacksonville is 892,062 (according to Google). By the probability in your survey 72% of them are football fans. So that gives us 642284.64 football fans (we're going to round to 642285 fans because well 0.64 fans is awkward).

So considering your reasoning the probability of the first person sampled being a football fan is: $$642285\over892062$$

The probability of the second person being a football fan is then: $$642284\over892061$$

If we multiply those together we get 0.51840035513 as opposed to 0.51840058112 from the original calculation. So the probabilities differ by 0.00004359372% which is pretty insignificant for an example problem.

So where does this become important: well let's say the population of Jacksonville is only 4 people and 3 of them are football fans. According to the textbook the answer for finding two football fans is $0.75^2$ or $0.5625$. While by your observation and the correct model it is $0.75*0.6666$ (3/4 * 2/3) or $0.5$. So now we have more than a 10% difference between the two answers.

As the population shrinks (either of football fans or of people in total) this consideration becomes ever more important. What's the probability of finding two football fans in a city with a population of 2 where 50% of them are football fans? Well the approach the example uses would give $0.5^2$ or $0.25$ while the correct approach gives 0 (there's only one football fan so you have 1/2*0/1).

What you have stumbled across is the difference of sampling with replacement vs sampling without replacement: https://web.ma.utexas.edu/users/parker/sampling/repl.htm

  • $\begingroup$ I understand your explanation. you've focused more on population size and sampling with replacement or without replacement. But they mentioned here because of persons are randomly selected that's why they are independent and we made this type of calculation. How? $\endgroup$ Oct 19, 2019 at 5:04
  • $\begingroup$ The wording is a little bit misleading in the question: They are independent draws from the same overall population (citizens of Jacksonville) and so they are statistically independent. Meaning the first person you pick has no influence on the second person that they pick (except through population effects as I mentioned in my answer). The way to combine independent probabilities is that you multiply them together. $\endgroup$
    – Patrick
    Oct 19, 2019 at 13:31

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