A friend of yours asserted at lunch today that, according to...

A friend of yours asserted at lunch today that, according to the weather forecast for tomorrow, there is a $52 \%$ chance of rain and a $60 \%$ chance of snow. "But that's impossible!" you blurted out, "the percentages add up to more than $100 \% . "$ Explain why you were wrong.

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Key Concepts

Mutually Exclusive Events

In probability, events are considered mutually exclusive when they cannot occur at the same time. In this context, rain and snow are not mutually exclusive because it is possible for both to occur during the same day. This means the probabilities of rain and snow do not need to add up to 100% as they are not the only possible outcomes of the weather event.

Addition Rule for Probabilities

The addition rule in probability states that for any two events, the probability of at least one occurring is the sum of their individual probabilities minus the probability that they occur together. Since there can be an overlap where it both rains and snows, simply adding the percentages without considering their intersection is incorrect.

Interpreting Forecast Probabilities

Weather forecast percentages often represent the chance of each type of weather event occurring, not mutually exclusive exclusive outcomes. Therefore, when forecasts indicate a possibility for both rain and snow, the percentages do not have to sum to any specific total, as these represent independent assessments of weather events that can overlap.

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