A broadcast meteorologist gives the following forecast:

Monday: 20% chance of rain
Tuesday: 20% chance of rain
Wednesday: 20% chance of rain
Thursday: 20% chance of rain
Friday: 20% chance of rain

A viewer is having a week long outdoor event that lasts from Monday through Friday. Monday morning the viewer asks the broadcast meteorologist what the chance for rain is for the entire week as a whole. In other words the viewer wants to know what the chance is it will rain on either Monday, Tuesday, Wednesday, Thursday, or Friday.

What is the answer? Assume the probability of precipitation (POP) is independent for each day and the forecasted POP does not vary with time.

Here are possible answers:

a) 20%
b) 33%
c) 50%
d) 67%
e) 92%
f) 100%

After deciding what the answer is, check below for the solution to this brain teaser.

SOLUTION: This situation represents the probability of rain within a 5 day period given at the beginning of the week and assumes each day is an independent probability. Two of the answer choices can be eliminated through applicable logic. It is known the probability of rain during the week is greater than 20% since each day has at least a 20% chance. It is also known the probability can not be 100% because the possibility is clearly evident that it might not rain at all during the week.

A probability for this case is solved by multiplying the probability it will not rain each day and subtract this from 100%. The left over value is the chance that is will rain during the week. The chance of no rain each day is 80%. Thus the chance for no rain each day put together is: 0.8 * 0.8 * 0.8 * 0.8 * 0.8 = 0.33 or 33%. Since the chance for no rain is 33%, the chance for rain is 100% - 33% = 67%.

(1 - 0.8^5) = 1 - 0.33 = 0.67 * 100% = 67%

Thus, there is a 67% chance (or a 2 in 3 chance) that the viewer will have rain sometime during the week.

**note: forecasts such as probability of precipitation occurring in a week are not practical in many situations since forecasted probability of precipitation generally does not remain constant during the week. This question represents the best guess probability given the forecast data currently at hand at the start of the week.