The sum of all probabilities is always equal to 1. This is a fundamental rule in probability theory and is known as the Law of Total Probability.

Explanation:

When we talk about probabilities, we are essentially talking about the likelihood or chance of an event occurring. Probability is always expressed as a number between 0 and 1, where 0 means that the event is impossible and 1 means that the event is certain.

For example, if we toss a fair coin, the probability of getting heads is 0.5 (or 50%) and the probability of getting tails is also 0.5 (or 50%). Together, these probabilities add up to 1.

The Law of Total Probability states that if we have a set of mutually exclusive and exhaustive events (i.e., events that cannot occur at the same time and that cover all possible outcomes), then the sum of the probabilities of these events is equal to 1.

For example, if we roll a fair six-sided die, there are six possible outcomes: 1, 2, 3, 4, 5, or 6. Each of these outcomes is mutually exclusive (i.e., we can only get one of them at a time) and exhaustive (i.e., they cover all possible outcomes). Therefore, the sum of the probabilities of these outcomes is 1:

P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1

In this case, each outcome has an equal probability of 1/6, so we can rewrite the equation as:

1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1

Which simplifies to:

6/6 = 1

This shows that the sum of all probabilities is always equal to 1, regardless of the number of events or their probabilities.