Probability that it rains at least one day of the work-week

Question

The probability of rain is $\frac{1}{2}$ for every day next week. What is the chance that it rains on at least one day during the workweek (Monday through Friday)?

Now, P(at least one)=1-P(none) The way I did it:

Probability that it does not rain from Monday to Friday : $\frac{1}{2^5}$

Probability it does not rain on both Saturday and Sunday: $\frac{1}{4}$

Probability it rains on either Saturday or Sunday or both: $\frac{3}{4}$

So we have, $1-(\frac{1}{2^5}*\frac{1}{4}$ +$\frac{1}{2^5}*\frac{3}{4})$

But the answer given is $1-\frac{1}{2^5}$. Basically this answer is not accounting for Saturday and Sunday. How can this be right?

Source:Manhattan Prep

Answer 1

The set-up is equivalent to

$X$ number of heads are obtained when a coin is tossed $n$ times , when the probability of obtaining head is "p". I hope you know that, X follows a binomial distribution with parameters "n" and "p".

As according to your question, we have, $n = 5,x=1,p=\frac{1}{2}=(1-p)$

$$Pr[X \geq x ] = Pr[X \geq 1 ] = 1 - Pr[X=0] = 1 - \frac{1}{2^5} $$

Answer 2

The question may be restated as "What is the probability that it will rain on at least one of Monday, Tuesday, Wednesday, Thursday, Friday". As such, the information whether it rains on the weekend is irrelevant for the question. The probability that it rains on none of the five weekdays is $\frac{1}{2^5}$. Thus, the final answer ist $1-\frac{1}{2^5}$.

Answer 3

We do not care at all what happens on the weekends so you can ignore them. If we want to account for them anyway, there are four possible weeks that meet our requirements of the complementary event.

So the probability that it does not rain on any of the weekdays, and rains on both Saturday and Sunday is (1/2^7).

The probability that it does not rain on any of the weekdays, and rains on Saturday, but not Sunday, yet again (1/2^7).

The probability that it does not rain on any of the weekdays, and rains on Sunday, but not Saturday, yet again (1/2^7).

The probability that there is no rain all week is once again (1/2^7).

Thus the probability of the complementary event occuring is (1/2^7)+(1/2^7)+(1/2^7)+(1/2^7) = 4(1/2^7) = (1/2^5).

Leaving us with a probability of 1-(1/2^5).

But again, we do not need to do this.

Answer 4

In this question we have to take days from Monday to Friday only. As work week is from Monday to Friday.

Because for Saturday Sunday we don't have any information. Maybe on both these days at home. And outside rain has no affect on his work.

So your answer without taking Saturday, Sunday is correct.