1. More Probability Concepts
Now let's continue building your foundation with some more basic probability concepts.
2. Conditional Probability
The conditional probability of an event A given event B is defined as the probability of A occurring given that B has already occurred. It's calculated as the probability of A intersection B divided by the probability of B.
3. Conditional Probability
Similarly, the probability of B given A is the probability of B intersection A divided by probability of A. Here we are calculating the probability of B occurring given that A has already occurred. Notice that the numerator in both these cases is identical. If we assume that neither P(A) nor P(B) is zero, then we can easily derive Bayes rule.
4. Bayes Rule
Bayes rule, expressed using conditional probabilities is shown in this equation. Although we won't go into much detail, you should know that it is an immensely popular rule for understanding the probability of an event using prior knowledge about factors that might have influenced that event.
5. Independent Events
Independent events are events where the probability of one occurring is independent of the probability of the other. This is expressed mathematically as the probability of A intersection B equals the product of the marginal probabilities of A and B. Consider two tosses of a coin. Seeing heads on the first toss is independent of seeing heads on the second toss. However, seeing heads on the first toss is not independent of seeing two heads in a row.
One nice outcome is that for independent events A and B, the conditional probability of event A given event B is the same as the marginal probability of event A. Let's work through a simple example.
6. Solar Panels & Clean Vehicles
Consider a neighborhood with 150 houses. You're given data regarding whether or not each house has solar panels installed and whether or not the owners have a clean vehicle - hybrid or an electric car. We want to find the probability of having solar panels conditional on the owners having a hybrid or an electric car. We first need to calculate the marginal probabilities.
7. Solar Panels & Clean Vehicles
The marginal probability of solar panels is sum of probabilities of solar with and without a clean vehicle, which is 40 divided by 150. Similarly we can calculate the other marginal probabilities.
The marginal probabilities have been calculated in the outermost row and column. Take some time to make sure you understand how they are calculated.
8. Solar Panels & Clean Vehicles
Finally, we want the probability of solar panels conditional on owners having a clean vehicle. Looking at the formula, the numerator is 30 divided by 150 and the denominator is 80 divided by 150, giving us an answer of 0.375. Try to work out the probability of having a solar panel without owning a clean vehicle and see if your answer is 1 divided by 7.
9. Let's practice!
Now let's work through some more challenging examples.