Question

If \(A\) and \(B\) are mutually exelusive events, \(P(A \cup B)=0,6, P(B)=0.4\), then \(P(A)=\ldots\)

Short answer

The probability of event \(A\) is 0.2.

Step-by-step solution

Understanding Mutually Exclusive Events

Two events, \(A\) and \(B\), are mutually exclusive when they cannot occur simultaneously. This implies \(P(A \cap B) = 0\).

Formula for the Union of Two Events

The probability of either event \(A\) or event \(B\) happening, denoted as \(P(A \cup B)\), is given by the formula: \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\] Since \(A\) and \(B\) are mutually exclusive, \(P(A \cap B) = 0\). Thus, the formula simplifies to: \[P(A \cup B) = P(A) + P(B)\]

Substituting the Given Values

We are given \(P(A \cup B) = 0.6\) and \(P(B) = 0.4\). We can substitute these values into the simplified formula: \[0.6 = P(A) + 0.4\]

Finding the Probability of A

To solve for \(P(A)\), subtract \(P(B)\) from \(P(A \cup B)\): \[P(A) = 0.6 - 0.4 = 0.2\] Thus, the probability of event \(A\) is 0.2.

Key concepts

Mutually Exclusive Events

Mutually exclusive events are quite an interesting concept in probability. Two events are considered mutually exclusive if the occurrence of one event means the other cannot possibly occur. Think of it like choosing between wearing either a red hat or a blue hat at the same time — you can only wear one. In terms of probability, this condition is represented as:

• \(P(A \cap B) = 0\)

This means there is no overlap, no seizing of simultaneous outcomes between the events. Understanding this concept is vital because it significantly simplifies the calculations required to determine the probability of various scenarios.

Probability of Union

The probability of union involves calculating the likelihood of either one or both of two events happening. When dealing with events that are not mutually exclusive, the formula is:

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

However, due to the joy of mutually exclusive events — which cannot happen at the same time as noted previously — the equation simplifies beautifully to:

• \(P(A \cup B) = P(A) + P(B)\)

Hence, the probability of at least one of the events occurring is just the sum of their individual probabilities. This simplifies our calculations greatly, sparing the need for overlaps.

Probability Formula

The probability formula plays a crucial role in determining the likelihood of particular outcomes. Probability essentially measures how likely it is for something to happen, expressed as a number between 0 and 1, where 0 means impossible and 1 means certain. In scenarios where events are mutually exclusive, calculating these probabilities becomes straightforward.
To find the probability of event \(A\) in this context, one can rearrange the formula for the union of events:

• \(P(A) = P(A \cup B) - P(B)\)

Given the problems in the exercise, where \(P(A \cup B) = 0.6\) and \(P(B) = 0.4\), it simplifies calculation to simply solve:

• \(P(A) = 0.6 - 0.4\)
• \(P(A) = 0.2\)

Thus, event \(A\) has a probability of 0.2 in this mutually exclusive scenario.