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How to Find the "Or" in Probability?

Published in Probability Calculation 3 mins read

To find the probability of event A or event B occurring, you typically use the following formula, which accounts for the possibility of overlap between the events:

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

Here's a breakdown:

  • P(A or B): This represents the probability of either event A occurring, event B occurring, or both occurring.

  • P(A): This is the probability of event A occurring.

  • P(B): This is the probability of event B occurring.

  • P(A and B): This is the probability of both event A and event B occurring simultaneously. This is crucial for avoiding double-counting when A and B can happen at the same time. Subtracting this accounts for the intersection of the events.

Why subtract P(A and B)?

When you add P(A) and P(B), you're counting the outcomes where both A and B occur twice. To correct for this overcounting, you need to subtract the probability of both events occurring together, P(A and B), once.

Example:

Suppose you roll a fair six-sided die.

  • Event A: Rolling an even number (2, 4, or 6)

  • Event B: Rolling a number greater than 3 (4, 5, or 6)

  • P(A) = 3/6 = 1/2

  • P(B) = 3/6 = 1/2

  • P(A and B): Rolling a number that is both even and greater than 3. The numbers 4 and 6 satisfy this, so P(A and B) = 2/6 = 1/3

Therefore, the probability of rolling an even number or a number greater than 3 is:

P(A or B) = P(A) + P(B) - P(A and B) = (1/2) + (1/2) - (1/3) = 1 - (1/3) = 2/3

Mutually Exclusive Events:

If events A and B are mutually exclusive (meaning they cannot both occur at the same time), then P(A and B) = 0. In this special case, the formula simplifies to:

P(A or B) = P(A) + P(B)

For example:

  • Event A: Rolling a 1
  • Event B: Rolling a 6

You cannot roll both a 1 and a 6 simultaneously. These are mutually exclusive. Therefore, P(A and B) = 0, and:

P(A or B) = P(A) + P(B) = (1/6) + (1/6) = 1/3

In summary, to find the "or" probability, use the formula P(A or B) = P(A) + P(B) - P(A and B), remembering to account for any overlap between the events. If the events are mutually exclusive, simply add their individual probabilities.

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