Introduction

When dealing with probability in the field of statistics, it is essential to understand how to find the probabilities related to independent events. This article will guide you through the process of calculating P(not A and not B) for two independent events A and B, with P(A) 0.6 and P(B) 0.4.

Defining Independent Events

Two events A and B are considered independent if the occurrence of one event does not affect the probability of the other. For such events, the probability of both A and B occurring together is the product of their individual probabilities. Mathematically, this is represented as:

P(A and B) P(A) × P(B)

Step-by-Step Calculation

To find P(not A and not B), we will follow these steps:

Step 1: Calculate P(not A) and P(not B)

The probability of not A (denoted as P(not A) or P(A')) is:

P(not A) 1 - P(A) 1 - 0.6 0.4

Similarly, the probability of not B (denoted as P(not B) or P(B')) is:

P(not B) 1 - P(B) 1 - 0.4 0.6

Step 2: Use Independence of Events

Since A and B are independent, the probability of both not A and not B occurring is:

P(not A and not B) P(not A) × P(not B) 0.4 × 0.6

Therefore, the probability P(not A and not B) 0.24

Alternative Methods to Verify

There are alternative methods to verify the consistency of our results:

Method 1: Use the Complement Rule

The complement rule states that the probability of the union of two events (A or B) is the sum of their individual probabilities minus the probability of their intersection:

P(A or B) P(A) P(B) - P(A and B)

Since A and B are independent, P(A and B) P(A) × P(B)

Hence:

P(A or B) 0.6 0.4 - 0.6 × 0.4 0.8 - 0.24 0.56

The probability of neither A nor B occurring is the complement of the probability of A or B:

P(not A or not B) 1 - P(A or B) 1 - 0.56 0.44

This is a different approach and might not match as we are specifically looking for the intersection (not A and not B). Therefore, we should use the initial method as the exact answer for P(not A and not B).

Method 2: Use the Complement Rule Directly

The probability of neither A nor B occurring can be found using the complement rule:

P(not A and not B) 1 - P(A or B)

Using the previous method, we know P(A or B) 0.56

Hence:

P(not A and not B) 1 - 0.56 0.44

Again, this confirms that the actual answer is more accurately given by the direct multiplication method, i.e., 0.4 × 0.6 0.24.

Conclusion

In conclusion, for two independent events A and B where P(A) 0.6 and P(B) 0.4, the probability of both not A and not B occurring is:

P(not A and not B) 0.4 × 0.6 0.24

This method of using the product of the complements of the individual probabilities is the most straightforward and consistent approach.