Classical Approach to Probability (Theoretical Probability)
The classical approach can only be used if each outcome has equal probability.
Empirical Probability (Experimental or Relative Frequency Probability)
The experiment is performed many times and the number of times that event A occurs is recorded. Then the probability is approximated by finding the relative frequency.
Notice that as n increased, the relative frequency seems to approach a number; it looks like it is approaching 0.163. You can say that the probability of getting a 4 is approximately 0.163. If you want more accuracy, then increase n even more by rolling the die more times.
These probabilities are called experimental probabilities since they are found by actually doing the experiment or simulation. They come about from the relative frequencies and give an approximation of the true probability.
The approximate probability of an event , notated as , is
For the event of getting a 4, the probability would be P(Roll a 4) = = 0.163
“‘What was that voice?’ shouted Arthur.
‘I don't know,’ yelled Ford, ‘I don't know. It sounded like a measurement of probability.’
‘Probability? What do you mean?’
‘Probability. You know, like two to one, three to one, five to four against. It said two to the power of one hundred thousand to one against. That's pretty improbable you know.’”
(Adams, 2002)
Figure 4-4 shows a graph of experimental probabilities as n gets larger and larger. The dashed yellow line is the theoretical probability of rolling a four of 1/6 0.1667. Note the x-axis is in a log scale.
# Roll a die 1,000 times and track the running relative frequency of a 4
set.seed(1)
rolls <- sample(1:6, 1000, replace = TRUE)
relfreq <- cumsum(rolls == 4) / seq_along(rolls)
relfreq[c(10, 50, 100, 500, 1000)] # the book's own rolls (Figure 4-3) gave 0.2, 0.12, 0.18, 0.162, 0.163
plot(relfreq, type = "l", log = "x", xlab = "n (log scale)",
ylab = "Relative frequency of a 4")
abline(h = 1/6, lty = 2, col = "red") # theoretical probability 1/6 = 0.1667
# Change the seed or the number of rolls -- every run drifts toward 1/6
Note that the more times you roll the die, the closer the experimental probability gets to the theoretical probability.
Figure 4-4
You can compute experimental probabilities whenever it is not possible to calculate probabilities using other means. An example is if you want to find the probability that a family has 5 children, you would have to actually look at many families, and count how many have 5 children. Then you could calculate the probability. Another example is if you want to figure out if a die is fair. You would have to roll the die many times and count how often each side comes up. Make sure you repeat an experiment many times, because otherwise you will not be able to estimate the true probability of 5 children. This is due to the law of large numbers, since the more times we repeat the experiment, the closer the experimental probabilities will get to the theoretical probabilities. For difficult theoretical probabilities, we can run computer simulations that can run an experiment repeatedly many times very quickly and come up with accurate estimates of the theoretical probability.
3. Subjective Probability
The probability of event A is estimated using previous knowledge and is someone’s opinion.
Adapted from Mostly Harmless Statistics by Rachel Webb (Portland State University), hosted on LibreTexts (stats.libretexts.org) and licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.
