An example of a bull market

Assume the probability of an upswing in each period
is and that market movements in different periods are independent. Then,
the following observation is made over the two-period model:

Find the probability that there is at least one
upswing.

**Solution**

The event that contains at least one upswing
is *A* = {(*u*, *d*), (*d*, *u*), (*u*, *u*)}. The only event that has all downswings is
{(*d*, *d*)}; hence:

This example illustrates the situation of a bull market, i.e. the probability of an upswing in each period is greater than

Now assume that the probability of an upswing in each period is , and that market movements in different periods are independent. This is known as a bear market.

Find the probability that there is at least one downswing over periods of time.

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