Based on weather data collected in Racine, Wisconsin, on Christmas Day, the weather had the following distribution:

Event Relative Frequency
Clear & dry 0.20
Cloudy & dry 0.30
Rain 0.40
Snow 0.10

Based on the data, what is the probability that next Christmas will be Rainy OR Cloudy & Dry?

Question

Based on weather data collected in Racine, Wisconsin, on Christmas Day, the weather had the following distribution:

Event                	      Relative Frequency     
Clear & dry	                0.20
Cloudy & dry	                0.30
Rain	                0.40
Snow	                0.10

Based on the data, what is the probability that next Christmas will be Rainy OR Cloudy & Dry?

BlackTom AI · Accepted Answer

To approach this problem, I’ll first identify the events and their given probabilities from the data.
- Rain has a relative frequency (probability) of 0.40.
- Cloudy & Dry has a relative frequency (probability) of 0.30.
- The events listed (Clear & Dry, Cloudy & Dry, Rain, Snow) are mutually exclusive outcomes for Christmas Day, so their probabilities add up to 1 when considered together.

Now I’ll evaluate each answer option:

- Option A: 0.4
  This value corresponds to the probability of Rain alone. It ignores the possibility that Cloudy & Dry could also occur. Since we’re asked for Rain OR Cloudy & Dry, including both outcomes is essential, so 0.4 is incomplete.

- Option B: 0.7
  This is the sum of the probabilities of Rain (0.40) and Cloudy & Dry (0.30). Because these two events are mutually exclusive, the probability of Rain OR Cloudy & Dry is the sum of their probabilities: 0.40 + 0.30 = 0.70. This aligns with the standard rule P(A ∪ B) = P(A) + P(B) when A and B are disjoint.

- Option C: 0.3
  This value matches only Cloudy & Dry in isolation. It neglects Rain entirely, so it cannot represent the combined probability of either Rain or Cloudy & Dry.

- Option D: 0.12
  This equals 0.40 × 0.30 if one were mistakenly attempting to multiply probabilities, which would be incorrect here because the event Rain OR Cloudy & Dry is about union (either outcome occurs), not the product of their probabilities. The product rule is used for independent joint events, not for union in this context.

In summary, the correct approach is to add the probabilities of Rain and Cloudy & Dry since they are distinct, mutually exclusive categories, giving 0.70. The other options either reflect only one of the events, attempt an inappropriate multiplication, or misinterpret the union of the events.

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