So what is the probability of tossing 3 heads in a row flipping a coin? Well another probability rule is called the joint probability rule. For independent events (that is one event does not affect the probability of the other), the rule is

P(A and B) = P(A) × P(B)

The result of flipping a coin does not affect the next flip of the coin, so these would be independent events and we can use this rule. The probability of flipping a heads is 0.5, so the probability of flipping three heads in a row is

P(flipping 3 heads) = 0.5 × 0.5 × 0.5 = 0.125 or 12.5%

Now let’s look at another experiment. Suppose you have a bag of 20 marbles, 10 blue ones and 10 red ones. The probability of picking a red marble is 10/20 or 0.5. If you replace the marble, shake the bag and redo the experiment, the probability of picking a red marble is still the same, that is the two experiments would be independent. So the probability of picking two red marbles in a row this way is 0.5 × 0.5 = 0.25. But what if you did not replace the marble? Before the second pick, the bag now has 19 marbles, 9 red and 10 blue so the probabilities of the second pick are affected by the first pick. This means that the two events are dependent.

If two events are dependent, say A depends on B, the way to show this is P(A|B). This means what is the probability that A occurs given that B occurred.

For dependent events, the joint probability rule is modified slightly:

P(A and B) = P(A|B) × P(B)

So you still just multiply the probabilities, but you must adjust the probability of A if B occurs.

Now back to the marble experiment without replacing the marble. What is the probability of picking two red marbles in a row?

Well for the first pick, we already know that the probability is 0.5. But for the second pick, the probability is 9/19 because there are only 9 red marbles now and a total of 19 marbles. So the probability of picking two red marbles without replacement is

P(2 red marbles) = P(second red marble|first marble is red) × P( first red marble) = 9/19 × 0.5 = 0.237. So the probability is slightly less picking two red marbles without replacement than it is with replacement.

This sets us up to do much more complex probabilities. In my next post, I’ll discuss probability trees.