# What is the population and sample in the coin flip experiment?

As a new learner of statistics I have some basic questions. In the experiment of coin flipping, if the coin is tossed for n times and it has been replicated over k trials that means the n-coin-flip has been repeated for k times, then what is considered to be the population and sample of our experiment?

Thanks

• The sample is what you have. The population is what you might have got: it is hypothetical and infinite. – Nick Cox Jan 21 '20 at 21:50
• How is the population infinite? The population or sample space in tossing 2 coins is: {HH, HT, TH, TT}. In tossing n coins, we can similarly specify the population. He did not ask about tossing an infinite number of coins nor did he ask how many tosses it takes to get the first Tails, in which case I'd agree with you, but as is, I think you're leading him astray. – ColorStatistics Jul 19 '20 at 17:38
• @ColorStatistics Why do you say that population and sample space are the same? – Dave Jul 19 '20 at 18:24
• @Dave: Damodaran N. Gujarati in his book "Basic Econometrics" on page 870 states "The set of all possible outcomes of a random, or chance, experiment is called the population, or sample space..." – ColorStatistics Jul 19 '20 at 18:27
• @Dave What ColorStatistics has explained is correct. Note that he is talking about "sample space" and not "sample". Informally we can say "sample space" is the set of different "samples". – Coder Jul 20 '20 at 18:53

Let's let your n=2, that is, we'll toss 2 coins. k is irrelevant to specifying the population, so we'll leave it aside for now.

The experiment of interest is tossing 2 coins. The population or sample space is the set of all the possible outcomes. An outcome is a complete specification of what could possibly happen when you conduct your experiment. So when you toss 2 coins, the population or sample space is the set of 4 possible outcomes {HH, HT, TH, TT}.

Now, let's repeat the experiment k=5 times. You might observe the following outcomes on these 5 flips of 2 coins: HH, HH, HT, TT, HT. This is your sample. Based on this sample of size 5 you can calculate the relative frequency with which each outcome occurred by dividing the frequency with which each outcome occurred by the total number of trials. As you increase k from 5 to 50, to 100, to 200, to 1000, to 100,000 and beyond you will notice that the relative frequencies of the 4 possible outcomes will approach more and more certain values. In the case of both coins being unbiased, you'll notice as you increase the number of trials that the relative frequencies will approach the theoretical probabilities of each of the 4 outcomes = (0.5)*(0.5)=0.25.

• Thanks for your time. Based on your answer and comments, I am quite sure you are well knowledgeable in statistics however, a misunderstanding has happened here I am afraid. n is not the number of coins in my question, we always have just one coin. For example, we toss a coin for 100 times, so n=100 then we replicate the 100-flips over 50 times, therefore k=50. In this case, population or sample space is {H,T} and the number of different samples is k=50 where the size of each sample is n=100. Thanks again for sharing your knowledge. – Coder Jul 20 '20 at 19:17
• As the clarification about the different number of replications: We perform like that to gain the mean number of heads to be used in the further statistical analysis. – Coder Jul 20 '20 at 19:24

The notion of a population does not really apply to coin tosses, or a lot of other random experiments. The population is a concrete way to introduce the difference between a sample from a distribution and the distribution itself in introductory statistics courses. Lots of people carry the term population over to discussions of distributions. We often speak of population moments, but we are referring to the moments of a distribution. There is no actual population.

The population is not the same as the sample space. The sample space is the set of possible outcomes. There is no implied relative frequency of occurrence. For a coin toss, the sample space is {Heads,Tails}. For a fair coin, both outcomes have equal probability. For an unfair coin, which has the same sample space, one of the two outcomes will be more likely.

• Keep in mind that population is the same as sample space. When you're flipping coins you are interested in some event, which as you might know, is a subset of the sample space. If there is no population in coin flipping then there is no sample space, and no subsets of the sample space so we would not be able to compute the probability of any event. – ColorStatistics Jul 19 '20 at 17:16