5. Chaining events together

Here is a simple probability model for the market during a bullish time period.

The market either goes up or down.

The probability that it goes up is 0.6.

The probability that it goes down is 0.4.

What happens one day has no effect on what happens the next.

The last statement, that today has no effect on tomorrow is vital here. It says that knowing what happened today gives you no information about what happens tomorrow.

This is the idea of independence.

If two events are independent then the probability that they both happen, is the probability that the first happens multiplied by the probability that the second happens.

What is the probability that the market goes up 2 days in a row?

This happens if it goes up on the first day and it goes up on the second. Therefore the answer is $0.6 \times 0.6 = 0.36$.

A more complicated question is: ``what's the probability we have less than 2 down days out of four?''

A good way of solving this is to construct a probability tree. It provides a systematic method for identifying all the possible outcomes. From it, you can identify the events of interest, and then sum their probabilities.

 

Figure 1: Probability tree for listing out the possible market outcomes over a four day period

Going back to the more complex question

What's the probability we have less than 2 down days out of four?

We can answer this by identifying the events in the tree that correspond to the event of interest. Less than 2 down days out of four, means either 0 or 1 down day.

These events can be identified at the bottom of the tree. They are those events with either 0 or 1 D.

Now we can simply add their individual probabilities to answer the question.

The key idea here, is that we get the probabilities at the bottom of the tree by multiplying the probabilities together in each branch.

We can multiply probabilities because the events are ``independent''. Knowing what happens on one day tells you nothing about what happens on the next.