Understanding Probability with Independent Events

The question asks for the probability of the event (A - B) when A and B are two independent events. The event (A - B) represents the outcomes where event A occurs, but event B does not occur. This is also known as the set difference between A and B, and can be written as A $\cap$ B', where B' is the complement of event B.

Probability of Set Difference (A - B)

For any two events A and B, the probability of the event (A - B) can be expressed using the formula related to intersections:

$$P(A - B) = P(A \cap B')$$

Alternatively, the probability of A occurring but B not occurring can be found by subtracting the probability of both A and B occurring from the probability of A occurring. This is because A can be partitioned into two disjoint events: (A $\cap$ B) and (A $\cap$ B').

So, we have:

$$P(A) = P(A \cap B) + P(A \cap B')$$

Rearranging this formula to find $P(A \cap B')$:

$$P(A \cap B') = P(A) - P(A \cap B)$$

Therefore, the probability of A - B is given by:

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

Leveraging Independent Events Property

The question states that events A and B are independent. This is a crucial piece of information. The definition of independence between two events A and B is that the occurrence of one event does not affect the probability of the occurrence of the other event.

Mathematically, for independent events A and B, the probability of their intersection (both A and B occurring) is equal to the product of their individual probabilities:

$$P(A \cap B) = P(A) \times P(B)$$

Calculating P(A - B) for Independent Events

Now we can substitute the property of independent events into the formula for $P(A - B)$:

We have the formula: $P(A - B) = P(A) - P(A \cap B)$

Substitute $P(A \cap B) = P(A)P(B)$ because A and B are independent:

$$P(A - B) = P(A) - P(A)P(B)$$

This expression gives the probability of event A occurring but event B not occurring, given that A and B are independent.

Comparing with Options

Let's compare our derived formula with the given options:

• Option 1: P(A) - P(A)P(B)
• Option 2: P(B) - P(A)
• Option 3: P(B) (1 - P(AB))
• Option 4: P(A) - P(B)

Our derived result, $P(A) - P(A)P(B)$, matches Option 1.

Let's briefly consider why other options are generally incorrect for independent events A and B:

• Option 2: P(B) - P(A) is generally not equal to P(A - B). P(A - B) relates to event A occurring without B.
• Option 3: P(B)(1 - P(AB)) is not a standard formula for P(A - B). $P(AB)$ is often written as $P(A \cap B)$, so this might imply $P(B)(1 - P(A \cap B))$. For independent events, $P(A \cap B) = P(A)P(B)$, so this becomes $P(B)(1 - P(A)P(B))$, which is not the derived result.
• Option 4: P(A) - P(B) is generally not equal to P(A - B). This is sometimes confused with P(A $\cup$ B) related formulas, but it is not correct for P(A - B).

Therefore, for independent events A and B, $P(A - B)$ is indeed equal to $P(A) - P(A)P(B)$.

Revision Table: Probability Concepts

Additional Information: Independent vs. Mutually Exclusive Events

It is important not to confuse independent events with mutually exclusive events. These are different concepts in probability.

• Independent Events: Two events A and B are independent if the occurrence of one does not affect the probability of the other. Mathematically, $P(A \cap B) = P(A)P(B)$. If P(A) > 0 and P(B) > 0, independent events can occur together.
• Mutually Exclusive Events: Two events A and B are mutually exclusive (or disjoint) if they cannot occur at the same time. Mathematically, $A \cap B = \emptyset$, which means $P(A \cap B) = 0$. If P(A) > 0 and P(B) > 0, mutually exclusive events are dependent, because if one occurs, the probability of the other occurring is 0.

The formula $P(A - B) = P(A) - P(A \cap B)$ applies to any two events. When the events are independent, we simply replace $P(A \cap B)$ with $P(A)P(B)$ to get the specific formula for independent events.