10.08 With or without replacement

 

With or without replacement?

One of the most difficult tasks in multi-stage experiments is to determine whether the subsequent stages of the experiment are dependent or independent, and so whether the probabilities assigned at each stage stay the same or change. 

One specific case you need to be aware of is that when we choose people (or things) from large populations. You may think that in the case of selecting people, we should not replace the first person before selecting the next. This would make the selections dependent and that we should change the probabilities. However, in practice, we realise that the probabilities will be changed in such an insignificant way that it will make only a minuscule difference to our final answer. And so, we consider these selections to be independent. 

Consider the following problem: a nationwide survey found that $64%$64% of people in a small country town have unreliable internet access.  If $3$3 people are selected at random, what is the probability that all three have unreliable internet access?

In this case, we haven't even been told how many people live in the town, so we assume that it is a large enough population that choosing $3$3 people will not affect the probabilities significantly. And so, we calculate our answer as 

$P(I)\times P(I)\times P(I)$P(I)×P(I)×P(I)$=$=$0.64\times0.64\times0.64=0.262144$0.64×0.64×0.64=0.262144 

 

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5