Question

A rainy day occurs once in every 10 days. Half of the rainy days produce rainbows. What percent of all days do not produce a rainbow?

• A. 95%
• B. 10%
• C. 50%
• D. 5%

Hint:

Multiply the chance of rain by the chance a rainy day gives a rainbow, then subtract that result from 100%.

Answer 1

The Correct Option is A

This question is about combining two separate chances, the chance that a day is rainy and the chance that a rainy day also brings a rainbow, into one overall chance, and then flipping that around to find the chance of no rainbow.

Out of every 10 days, 1 day is rainy. Out of those rainy days, only half go on to produce a rainbow. So out of every 10 days, the number of days with an actual rainbow is $\frac{1}{2} \times 1 = 0.5$ days. In percentage terms, that is $\frac{0.5}{10} \times 100 = 5\%$ of all days.

1. 95%: this is everything left over once the 5% rainbow days are removed from the full 100%, so it correctly represents the days without a rainbow.
2. 10%: this looks like the frequency of rainy days doubled, or a slip where someone forgets to take "half of the rainy days" into account; it does not represent the no rainbow days.
3. 50%: this would be the answer only if someone stopped at "half the rainy days produce rainbows" and mistook that for "half of all days", which is not what the question says.
4. 5%: this is the percentage of days that DO get a rainbow, the opposite of what the question asks for.

The correct value is 95%, since only 5% of all days end up with a rainbow, so the remaining 95% do not.

Let's summarize:

• Chance of rain: 1 in 10 days, or 10%.
• Chance of a rainbow: half of the rainy days, so 5% of all days.
• Chance of no rainbow: the complement of 5%, which is 95%.

So 95% of all days do not produce a rainbow, option (A).