1. What is the probability of at least breaking even?

2. What is the probability of achieving next year’s sales goal?

3. What additional information do you want to be more confident about your probability assessments in the previous two questions?

4. How would your answer for questions 1 and 2 differ if the mean shifted from 100,000 to 106,000?

1. What is the probability of at least breaking even?

The answer is that any number of units above 86, 538 (the break-even point) would be profitable for Pillai Company.

Now, let us use this break-even cost information to work out the probability of breaking even.

Take a look at the z Table at the end of this appendix. The arrow pointing to the left hand side of normal distribution shown earlier gives us the answer. Note that a negative z means this value is below the mean. Looking at the z Table at the line with 1.6 and along column .08, we get a value of 0.45352. Now add on the 0.5 of the right tail, and this gives us the overall probability. The answer is 0.95352. This means that we have a 95% probability of breaking even, resulting in little pressure on the manager to break even.

2. What is the probability of achieving next year’s sales goal?

See the next z Table and look along the line with 1.4 and column .01 that has a circle around it.

As z is positive, it is on the right-hand side of the curve.

If 0.5 is the right tail, then 0.5 - 0.42073. The answer is 0.08. This means that the manager has an 8% chance of achieving the target. This is quite a stretch target: If the manager does not achieve this business unit performance, what are the consequences?

3. What additional information do you want to be more confident about your probability assessments in the previous two questions?

a. How useful is the past for predicting the probability of the future? To address this point, a few issues need to be considered. First, if there is no change in technology, learning by employees, or substantive changes in business processes, the probability assessment is fairly robust. Second, using the past data is valid as it provides a useful range of points to plan and map out. Ignoring the past and forecasting does not give any indication of how robust or confident the estimation of probability is going to be.

b. Are last year’s sales being higher than the mean an indication of changing conditions? This is the key to setting targets. If this year’s sales are higher as the demand for products or services has grown, then clearly the demand function has shifted. However, last year’s sales being higher than average does not mean a shift in demand. As you look at a normal curve, last year’s sales of 106,000 units was just another point on the distribution. There has been no shift in demand.

c. What is the distribution of the fixed costs? Over the points of the normal curve, fixed costs should remain constant. In other words, the costs of capacity or fixed costs should not have changed across the distribution curve. If the fixed costs do change, then it is likely that there is a step function to these costs. In other words, if moving to a sales volume of 106,000 units meant leasing new warehouse space or new machines to manufacture this new capacity, then the cost analysis needs to be investigated further to understand the probability of breaking even.

If the distribution of fixed costs does not change, in other words, across the normal curve for sales, fixed costs remain constant, and the variable cost per unit remains constant (there are no economies or diseconomies of scale), the fundamental cost structure of doing business remains the same across the new level of 106,000 units.

4. How would your answer for questions 1 and 2 differ if the mean shifted from 100,000 to 106,000?

Now we are stating that last year’ sales of 106,000 has become the new mean. In other words, we have shifted the curve to the right to get the 106,000 as the peak, resulting in a 50% chance of getting last year’s sales as the future target.

Basically, a senior management has decided to increase (or ratchet) the sales forecast where 106,000 units are held to have a 50% likelihood of occurring. They will use the same analysis as the first two questions.

Computing Probability:

The probability is 0.49245. This means that the likelihood of achieving the break-even point is now higher and goes to 99.245% (.50 + 0.49245). Increasing performance targets can result in what is called ratcheting and is discussed in chapter 4.