## Transition to a Standard Curve - Another Example

Joe holds shares. The annual profit on these shares has a normal probability with an expectation of 9.6% and a standard deviation of 8%. This expectation of 9.6% profit does not guarantee a 9.6% profit each year.

The probability distribution of the profit can be represented by a bell-shaped curve:

Diagram

What are the chances that Joe will make a profit this year?

Earning a profit means obtaining a profit larger than zero (i.e., a loss is actually a negative profit). What are the chances that the profit this year will be greater than 0. We are actually looking for the area shaded in the following diagram:

Diagram

We must first find the standard unit.

The standard unit of zero is ^{(0-9.6)}/_{8} = ^{-9.6}/_{8} = -1.2. We add this to the diagram in the **standard units** row

Diagram

We have reiterated the original question as follows: What is the area to the right of –1.2 in the standard probability distribution?

In order to answer this question, we must examine the table. The area **to the left of** -1.2 is -0.1151, so the area **to the right** of -1.2 is 1 - 0.1151 = 0.8849. The chances that Joe will earn a profit this year are therefore 0.8849 (out of 1). In other words, the chances that Joe will earn a profit are 88.49%.