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Students usually learn a classic definition of probability early in the process of developing basic statistics. This definition states that, given the same probability of all outcomes or events in an experiment, the probability of a given event is equal to the number of favorable outcomes divided by the number of all possible outcomes.

The simplest example of a classic probability experiment is a coin toss: you toss a coin and see which side it lands on: heads (H) or tails (T). The sample space consists of two events {H,T}. Here, the coin is considered symmetric. According to the classic definition of probability, the odds of getting heads or tails are equal to 1/2. The classical probability formula is a natural concept and students pick up on it immediately.

Since the number of all possible outcomes in the coin toss experiment is small, one quickly moves on to a more complex coin toss experiment. Consider experiments involving flipping two coins. Most probability books present such experiments without further explanation. For example, in the classic book on probability*A first course in probability*, the following example is given:

If the experiment consists of tossing two coins, then the sample space consists of the following four points: S = {(H, H), (H, T), (T, H), (T, T)}

(Video) The coin flip conundrum - Po-Shen Loh

The phrase "flipping two coins" suggests that two coins are flipped at once. Is the reasoning in such an experiment really that simple? Not exactly. Students question the size of the sample space when they say we are tossing two identical coins.

## The problem

Consider the experiment of tossing two identical coins in one toss. A significant percentage of students agree that there are only three possible outcomes of this experiment. That is, students argue that you can have two heads (H, H), two tails (T, T) or one heads and one tails (H, T). Nothing unusual, as this is exactly what can be observed. When asked what the probability of getting heads and tails is, students answer 1/3 (a favorable one out of three possible outcomes), which is in line with the classic definition of probability. Such reasoning can be forced by posing the following ternary problem:

*Consider the following experiments, and for each experiment, write down all possible outcomes, count the number of outcomes, and determine the probability of getting heads and tails.*

**Attempt 1:**You flip a coin twice. Label all possible outcomes as ordered pairs (-, -), where the first entry is the result of the first toss and the second entry is the result of the second toss.

**Attempt 2:**In a single flip, you flip two coins of different types.

**Attempt 3:**You flip two identical coins in one toss.

All three experiments were deliberately framed as a problem so that students would suspect that their test rooms might not be the same. Because they were given instructions on how to label the results, almost all of the students were able to identify four possible outcomes of Experiment 1. Additionally, the students concluded that if there is agreement on the order in which the results are labeled, there are four possible outcomes in Experiment 1. 2. like.

In summary, the results of the first two experiments can be written as ordered pairs: (H, H), (H, T), (T, H) and (T, T). Using the classic definition of probability, the students concluded that the probability of getting heads and tails in both experiments is 2/4 = 1/2. Furthermore, more than 60% of students wrote that the three experiments have four outcomes and probabilities of getting heads and tails equal to 1/2. However, for almost 40% of the students, the answer was shown in Figure 1.

Figure 1: One of the solutions to the problem.

There were two groups of students confronted with differing opinions about the total number of outcomes in Experiment 3. The main issue was whether we needed to distinguish between experiments in which two different or identical coins are tossed in a single toss. Students were asked to explain why they thought the total number of outcomes in Experiment 3 was either three or four. They discussed the issue in small groups in the spirit of the peer-to-peer instruction presented by Eric Mazur in"Peer Instruction: Getting Students to Think in the Classroom".We do not get involved in the discussion.

In the end, the minority convinced the majority to change their minds! The reason is that there is no way to visually distinguish between the results (H,T) and (T,H), there are only three results that can be discerned. These are the H-H, H-T, and T-T results, and we can stop writing them as ordered pairs.

In Experiment 3, students concluded that there are only three outcomes and that the probability of getting heads and tails is 1/3 by the classical definition (one favorable outcome out of three possible outcomes). Almost all students agreed with this after the discussion.

Our goal of actively and authentically involving students in the problem has been achieved. Students took responsibility for solving the problem; It was time to put them where they should be.

## guide to the right conclusion

We didn't delve into the axiomatic definition of probability, so we introduced students to three aspects of the definition of probability. In addition to the classical definition, the probability of occurrence of an event can be seen as the relative frequency with which we expect the event to occur in a large number of trials. Furthermore, one can speak of a subjective concept of probability as a measure of conviction. In a single coin toss experiment, where there are two possible outcomes and with the logical assumption that the coin is symmetric, all three definitions agree. A subjective or epistemic interpretation of probability (such as the classical one) suggests that heads and tails are equally likely to occur, each with probability 1/2. The definition of relative frequency assumes that the experiment is repeated many times and the probability of an event is found as the bound of the resulting relationship between the number of occurrences of an event and the number of times the experiment is performed. When in doubt about a probability, the relative frequency approach is just a means of finding the correct answer.

The students concluded that there is a way to check whether the probability of getting heads and tails in two identical coin flips is 1/3. They experimented with tossing two identical coins several times. Students were divided into pairs and each pair counted the number of heads and tails that appeared when two identical coins were tossed 30 times. All results were pooled to calculate the relative frequency of the event.

The students were surprised when they discovered that it was almost half, not 1/3! This observation led to the following conclusion: although one cannot distinguish between H-T and T-H outcomes, they still occur.

In conclusion, it was found that the sample space in the experiment of tossing two identical coins consists of four outcomes. Also, the probability of getting heads and tails is twice as likely as getting two heads or two tails. We noticed that both those who got it right and those who didn't seem to be confident in the size of the pattern space.

## problem discussion

The toss of two coins is the most elementary random experiment mentioned in almost every statistics textbook. Most textbooks start with a simple understanding that there are a total of four possible outcomes in experiments involving flipping two coins. The authors do not comment on whether the two coins are identical.

Depending on the case, students use the standard room size to distinguish between flipping two different coins or two identical ones. We faced the same problem year after year, generation after generation: A relatively large percentage of students agreed that, in the experiment of tossing two identical coins, the sample space consisted of only three events. Even those who thought there were four possible outcomes weren't confident enough to defend their opinion when faced with another.

The previous lesson leading students to the correct conclusion about sample space size takes time that the teacher may lack. It would be best to rule out any misunderstandings about the rehearsal room right from the start. There is an easy way to do this: talk about a coin tossed twice. In such experiments, the results can legitimately be called ordered pairs, and students will likely have no trouble writing all four. This approach can also be found, for example, in some statistics books.*A modern introduction to probability and statistics: understanding why and how*.

We know from our classroom practice that students can easily generalize and calculate the number of all possible outcomes of the experiment of tossing a coin more than twice. So, in repeated experiments*norte*times the total number of results equals 2^{norte}🇧🇷 The only weakness of such experiments compared to several identical coin toss experiments is that, if actually performed, they would be more time consuming. However, there is a small price to pay, if any, as most of the experiments we "do" are done mentally (i.e., justifying likely outcomes and their probabilities without actually experimenting).

The reader may object to the idea of talking about flipping a coin twice, rather than flipping two identical coins at the same time. However, if we think about the results of the experiment of flipping a coin twice, we can better understand that they are the same as the experiment of flipping two coins at the same time. Regardless of whether two identical coins are thrown at the same time, they are unlikely to land at the same time. This justifies consideration of experiments on flipping a coin twice.