Question

Ella's college professors assign homework independently from each other. Ella knows that there is a 60% chance that she will have math homework tonight and a 70% chance that she will have English homework. What is the probability that she will NOT have math or English homework tonight?

10%

12%

42%

70%

Answer 1

Let \(M\) denote the event that math is assigned, and \(E\) the even that English is assigned.

You want to find \(P(M'\cup E')\) (assuming the "or" means the inclusive or), where \(M'\) and \(E'\) are the complements of \(M\) and \(E\) (i.e. math is not assigned, English is not assigned).

Answer 2

So \(P(M)=0.6~~\iff~~P(M')=0.4\), and \(P(E)=0.7~~\iff~~P(E')=0.3\).

\(M\) and \(E\) are independent events, which means \(P(M\cap E)=P(M)P(E)\). If two events are independent, then their complements are also independent.

Recall the inclusion/exclusion formula:
\[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]
Setting \(A=M'\) and \(B=E'\),
\[P(M'\cup E')=P(M')+P(E')-P(M'\cap E')\]
and since the complements are independent,
\[P(M'\cup E')=P(M')+P(E')-P(M')P(E')\]

Answer 3

Im sorry, What. o-o

Answer 4

Where do start getting confused?

Answer 5

Everywhere o-o.

Answer 6

Okay, start from the beginning. Do you follow my first comment? I'm just defining the events and showing what you want to find.

Answer 7

Yeah, i get that

Answer 8

Good, so you're *not* confused everywhere :P

The first line of the second comment: I'm just computing the complements' probabilities. This is using the property that \(P(A)+P(A')=1\) for an event \(A\) and its complement \(A'\). Surely you've learned about this one?

Answer 9

Nope. Not at all. ;-;

Answer 10

Okay, well it's not too hard to understand. The complement of an event \(A\) is the event that \(A\) does not happen. For example, if \(A\) is the event that it will rain tomorrow, \(A'\) refers to the event that it will NOT rain tomorrow.

If the probability of \(A\) is, say, 0.5, then the probability of \(A'\) is necessarily 0.5 as well. This is because it will either rain or it will not rain tomorrow. This event (the union) is denoted \(A\cup A'\). One of the events will occur, so the probability that *at least one of these events* occurs is 0.5+0.5=1. Does that make sense?

Answer 11

Yeah, yah just lost meh. so how do i find the probability that she didnt have math or english homework?

Answer 12

If you don't want to learn how to approach the problem, that's fine by me. I won't waste my time. But I also am not going to just give you the answer.

Answer 13

i dont just want the answer, youre confusing me. it just doesnt make sense to me. i cant pick up on something so quickly that i havent learned yet.