To find the expected time of an activity, we can use the formula from the Program Evaluation and Review Technique (PERT). PERT considers three time estimates for an activity: optimistic time ($t_o$), most likely time ($t_m$), and pessimistic time ($t_p$).

Understanding the PERT Expected Time Formula

The formula for calculating the expected time ($T_e$) of an activity in PERT is:

$$T_e = \frac{t_o + 4t_m + t_p}{6}$$

This formula gives more weight to the most likely time ($t_m$) compared to the optimistic ($t_o$) and pessimistic ($t_p$) times, reflecting its higher probability of occurrence.

Calculating the Expected Time for the Activity

We are given the following time estimates for the activity:

• Pessimistic time ($t_p$) = 14 days
• Most likely time ($t_m$) = 8 days
• Optimistic time ($t_o$) = 2 days

Now, let's substitute these values into the PERT expected time formula:

$$T_e = \frac{t_o + 4t_m + t_p}{6}$$

$$T_e = \frac{2 \text{ days} + 4 \times 8 \text{ days} + 14 \text{ days}}{6}$$

First, calculate $4 \times t_m$:

$$4 \times 8 = 32 \text{ days}$$

Next, add all the terms in the numerator:

$$2 + 32 + 14 = 48 \text{ days}$$

Finally, divide the sum by 6:

$$T_e = \frac{48 \text{ days}}{6}$$

$$T_e = 8 \text{ days}$$

Summary of Expected Time Calculation

Using the PERT formula with the provided optimistic, most likely, and pessimistic times, the calculated expected time for the activity is 8 days.

The steps were:

1. Identify the given optimistic time ($t_o$), most likely time ($t_m$), and pessimistic time ($t_p$).
2. Use the PERT formula: $T_e = (t_o + 4t_m + t_p) / 6$.
3. Substitute the given values into the formula.
4. Perform the calculation to find the expected time ($T_e$).

Based on the calculation, the expected time for the activity is 8 days.