Experimental Probability

$$\textbf{Experimental Probability}$$

Experimental probability, or empirical probability, is a type of probability based on experiments, observations, and real-world data. When you calculate the experimental probability of an event occurring in the future, you are taking into account how many times the event has already happened in the past. Experimental probability is different from Theoretical Probability, because it depends on experimental data to predict a future outcome, while theoretical probability only relies on the hypothetical amount of successful outcomes to predict a future outcome.

We can calculate experimental probability of an event, $E,$ occurring using the following formula:

Here, $P(E)$ represents the experimental probability that event $E$ will happen, $n(E)$ represents the number of times $E$ has already occurred, and $n(T)$ represents the total number of trials you have performed in your experiment.

Let's use this formula in an example!

We can tackle this problem in steps. Let $E$ be the event that the coin lands on tails. We are trying to calculate $P(E),$ the experimental probability that the coin will land on tails on the next flip.

First, let's figure out $n(T),$ the number of trials you have conducted. Since you have already flipped the coin $36$ times, you have conducted $36$ trials, and recorded the data from these trials. Thus $n(T) = 36.$

Now, let's find $n(E),$ the number of times event $E$ has occurred. You know that you landed on heads $21$ times out of the $36$ flips. Thus, you landed on tails $36-21=15$ times out of the $36$ flips. So, $n(E) = 15.$

Now that we have $n(E)$ and $n(T),$ we can substitute this information into our experimental probability formula!

$$\begin{align} P(E) &= \frac{n(E)}{n(T)}\\ \\ &= \frac{15}{36}\\ \\ &= \frac{5}{12} \end{align}$$

Thus, the experimental probability of our next coin flip landing on tails is $\frac{5}{12}.$ Notice that the experimental probability of the coin landing on tails is different than the theoretical probability of the same coin landing on tails, which is $\frac{1}{2}.$ Sometimes experimental and theoretical probability match up, and sometimes they don't!

Let's do one more example of how to calculate experimental probability.

Let's figure out $n(E)$ and $n(T)$ so we can substitute them into our experimental probability equation!

We drew a silver marble once, and then we repeated the process $31$ more times. Thus, we performed $1 + 31 = 32$ trials in total. So, $n(T) = 32.$

We drew a silver marble once, and then we drew a silver marble 13 more times. Thus, we drew a silver marble $1+13 = 14$ times in total. So, $n(E) = 14.$

Now we can substitute $n(E)$ and $n(T)$ into our experimental probability formula:

$$\begin{align} P(E) &= \frac{n(E)}{n(T)}\\ \\ &= \frac{14}{32}\\ \\ &= \frac{7}{16} \end{align}$$

Thus, the experimental probability of drawing silver on our next turn is $\frac{7}{16}.$

It's worth mentioning that we cannot calculate the theoretical probability of drawing silver. The formula for theoretical probability, $P_T(E) = \frac{\text{number of favorable outcomes}}{\text{number of possible outcomes}},$ depends on how many silvers there are in the bag (number of favorable outcomes) and the total amount of marbles in the bag (number of possible outcomes). Since we don't have this information, we cannot determine the theoretical probability of drawing silver on our next turn.

The history of Statistics begins, amusingly enough, with probability and gambling. Gambling in various forms has existed since antiquity, and like anything else, when there is money involved, people want to find new and inventive ways to get better at acquiring it.

Blackjack is a game which is relatively easy to win with some statistical knowledge.

One way to find out how to win, or at least find the probability of winning, is to sit and watch people and count how many times they win out of the times that they play.

This is the basis of simple Experimental Probability. Experimental probability is the observed probability of an event occurring. In most cases, this is found by observing the event (a game, for example) and seeing how many times the player wins (the number of "Successes") or the player loses (the number of "Failures").

The probability of an event occuring is generally: $$\frac{\text{Number of Successes}}{\text{Total Number of Events}}$$

So let's say you're playing a game in which you roll a six sided die repeatedly until you roll either a $1$ or a $6.$ If you roll a $6,$ you win a dollar, if you roll a $1,$ you lose a dollar. (This is a much simplified version of a game called Craps)

Given that the game is entirely random, you can watch and see how many times a person wins - in the twenty games you see, the person wins $9$ times. This means that your experimental probability of winning is $\frac{9}{20}$ or $.45.$ Which means that there is a $45\%$ chance of winning, as far as you can tell.

(It turns out that you can use logic to find the theoretical probability of this occurring - it is exactly $\frac{1}{2},$ as you might expect if you think about it long enough.)

(Video) Experimental vs. Theoretical Probability

by Nat Oliver

Nat Oliver has a great video explaining what the difference is between theoretical AND experimental probability. He gives examples and explains how to calculate both.