# Alternative formula for the Rule of Addition not working for some problems

• Juwane
In summary, the rule of addition in probability states that the probability of the union of two events is equal to the sum of their individual probabilities minus the probability of their intersection. However, there is an alternate formula that can be used when the two events are independent. In this case, the probability of the intersection is equal to the product of the individual probabilities. The alternate formula is not applicable if the events are not independent and the intersection must be determined separately.
Juwane
In probability, the rule of addition is:

$$P( A \cup B) = P(A) + P(B) - P(A \cap B)$$

This can be written as

$$P( A \cup B) = P(A) + P(B) - P(A)P(B)$$

However, I've seen that using both formulas for the same problem gives different answers in some problems. Does anyone know why? Is there a condition for applying the second formula?

Juwane said:
Is there a condition for applying the second formula?

Yes, the two must be independent. The first one works in all cases.

But consider this example:

A shopper goes to a fruit store. The probability that he will buy (1) bananas is 0.40, (2) oranges is 0.30, and (3) both bananas and oranges is 0.2. What is the probability that the shopper will buy bananas, oranges, or both?

Using the usual formula:

$$P( B \cup O) = P(B) + P(O) - P(B \cap O) = 0.40 + 0.30 - 0.2 = 0.50$$

Using the alternate formula:

$$P( B \cup O) = P(B) + P(O) - P(A)P(B) = 0.40 + 0.30 - (0.4)(0.3) = 0.58$$

Why they are different?

Juwane said:
Why they are different?

Because the probabilities are not independent. (Didn't you see my post?)

CRGreathouse said:
Because the probabilities are not independent. (Didn't you see my post?)

I did see your post. That's why I said, "But consider this example".

Anyway, when the probabilities are independent, we don't need the minus part of the formula. This is enough:

$$P( A \cup B) = P(A) + P(B)$$

Then what is the use of writing the formula in the alternate form?

That's disjoint, not independent.

Juwane said:
This is enough:

$$P( A \cup B) = P(A) + P(B)$$

Then what is the use of writing the formula in the alternate form?

Only if probabilities are independent does $$P(A)P(B)=P(A)\cap P(B)$$. Otherwise you must be given or otherwise determine the intersection.

EDIT: A zero value for the intersection of P(A) and P(B) is not consistent with independence for non zero P(A),P(B).

Last edited:

## 1. Why does the alternative formula for the Rule of Addition not work for certain problems?

The alternative formula for the Rule of Addition may not work for certain problems due to the complexity of the problem or the specific conditions of the problem. It is possible that the alternative formula is not applicable in these cases and the original Rule of Addition should be used instead.

## 2. How do we know when to use the alternative formula for the Rule of Addition?

The alternative formula for the Rule of Addition should be used when the problem involves mutually exclusive events, meaning that they cannot occur at the same time. It can also be used when the problem involves non-mutually exclusive events and the probability of both events occurring is known.

## 3. Can the alternative formula for the Rule of Addition be used for any type of problem?

No, the alternative formula for the Rule of Addition is only applicable for problems involving probability. It cannot be used for other types of mathematical problems.

## 4. What are the limitations of the alternative formula for the Rule of Addition?

The alternative formula for the Rule of Addition may not work for problems involving dependent events, where the outcome of one event affects the probability of the other event. It also cannot be used for problems with more than two events.

## 5. How can we verify that the alternative formula for the Rule of Addition is correct?

The alternative formula for the Rule of Addition can be verified by using it to solve a problem and comparing the results to the original Rule of Addition. It can also be verified by checking if the sum of the probabilities of all possible outcomes is equal to 1.

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