Probability gives us a way to numerically quantify the likelihood of something occurring. Some probabilities are second nature to us as we have seen them through the course of our lives. For example, we might know that the chances of heads landing face up when I toss a coin are 50%. Probability is one of the mist fundamental tools used in statistics and we will see it arise as we continue through the class.

Probabilities are reported as values between 0 and 1, but we often report them as percentages as percentages make more sense to the general population. A probability of zero means something cannot happen. A probability of 1 means something is guaranteed to happen. The closer my probability is to 1, the more likely the event is to occur. The close my probability is to zero, the less likely the event is to occur.

There are three main types of probabilities we see:

Classical Probability - Classical probability is also known as the "true" probability. We can compute classical probability as long as we have a well-defined sample space, and a well-defined event space. We compute the probability of an event E as follows: P(E)=n(E)n(S) when n(E) refers to the number of elements in our event space and n(S) refers to the number of elements in our sample space. For example, let's take a look at a six-sided die. We can define our sample space as all outcomes of the roll of the die, which gives us the following: S = {1,2,3,4,5,6}. If we let our event E be that the outcome is the number 3, our event space becomes the following: E = {3}. In order to compute the classical probability, we take the number of elements in our event space and divide it by the number of elements in our sample space. This example gives us P(E)=n(E)n(S)=1/6. So, the probability of rolling the number 3 will be 1/6.

Empirical Probability - Empirical probability, also known as statistical probability or relative frequency probability, is probability calculated from a set of data. We compute it the exact same way we computed relative frequency by taking the number of times our event occurred divided by the number of trials we ran. The formula is as follows: P(E)=FrequencyofEventETotalFrequency. Taking a look at the die example, we can run an experiment where we roll a die 20 times and count the number of times the value 3 shows up. Suppose we do this and see that in 20 rolls of a six-sided die, the number 3 shows up five times. We can compute the empirical probability as follows: P(E)=FrequencyofEventETotalFrequency=5/20=1/4.. We now see that based on our experiment, the probability of rolling a 3 was found to be .

The law of large numbers tells us this: As my sample size increases, the empirical probability of an event will approach the classical probability of the event. When we have smaller sample sizes, we can often see oddities arise. For example, it is entirely possible to flip a fair coin and see the value of heads arise 5 times in a row, or even 9 times out of 10. Our empirical probability is far off our classical probability at this point in time. However, if I continue to flip my coin, I will see my empirical probability starts to approach our classical probability value of 0.5.

Subjective Probability - Subjective probability comes from educated guess, based on past experiences with a topic. For example, a teacher might say that if a student completes all their Statistics assignments before the due date, the probability they pass the course is 0.95.

1. What was your proportion of heads found in Step 1 (Hint: To do this, take the number of heads you observed and divide it by the number of times you flipped the coin). What type of probability is this?
2. How many heads would you expect to see in this experiment of 10 coin flips?
3. What was your proportion of heads found in Step 2 (Hint: To do this, take the number of heads you observed and divide it by the number of times you flipped the coin) What type of probability is this?
4. How many heads would you expect to see in this experiment of 20 coin flips?
5. your proportions differ between our set of 10 flips and our set of 20 flips? Which is closer to what we expect to see?