00:01
All right, your question starts with a pair of dice being tossed.
00:05
First thing i'm going to do is lay out a sample space for all possible outcomes, and then we'll move on to your questions.
00:16
Okay, so this sample space is going to represent the sum that we could get, which is two.
00:20
If we had a one and a one, three, four, five, six, and seven.
00:26
And all i'm doing was adding this one to every roll up here.
00:31
Now i'll just keep moving down and adding two plus one, two plus two, and so on.
00:38
So that's going to be three, four, five, six, seven, eight, four, five, six, seven, eight, nine, five, six, seven, eight, ten, seven, nine, ten, eleven, and seven, eight, ten, eleven, and seven, eight, ten, eleven, and seven, nine, ten, and seven, and seven, 12.
01:05
All right, so that's representing all the possible sums that we can get.
01:12
Now, event a, i'm sorry, event e is the first one they want, and we're supposed to list all elements of e.
01:20
Let e be the event that the sum is six.
01:23
So here's your sums that are six.
01:27
Now i think what we want to do is list out the rolls of the die that create those sixes.
01:35
So we could roll.
01:36
A 5 then a 1 that's this first one we could roll a 4 then a 2 we could roll a 3 and a 3 we could roll a 2 and a 5 i'm sorry 2 and a 4 or we could roll a 1 and then a 5 that represents all outcomes that lead to a sum of 6 now event a is saying let a be the event that the two appears on at least one die.
02:18
So we're going to have two and a one, a two and a two, a two and a three, two and a four, two and a five, two and a six.
02:31
We could also have one and a two.
02:34
I don't need to write two and a two again, because i don't want to count that.
02:39
That's the same thing.
02:42
So we don't want to write that, but we can have a three then a two, a four then a two, a five then a two, and a six then a two.
02:58
So i believe that's what they're asking for when they say list the elements in each event.
03:04
Are the events independent, dependent, or mutually exclusive? to answer whether they're independent, dependent, or mutually exclusive, i think it would be best for us to first find the answers to the probabilities that they're asking.
03:20
So we'll come back to the independent, dependent.
03:23
Mutually exclusive question.
03:26
The probability of e occurring, there are five outcomes that satisfy event e out of 36.
03:36
And i'll just leave it 536.
03:39
A lot of times you want a fraction for this answer.
03:44
Letter b is asking for the probability of event a.
03:53
And event a has a total of 11 outcomes that satisfy it out of 36.
04:02
So you can count these up.
04:04
There's 11 there, 11 different groups.
04:09
Now part c is saying, what is the probability of event a given that e occurred? okay, the way we calculate that is the probability of event a and e occurring, divided by the probability of e alone.
04:31
Now, a and e occurring means that we have to have a sum of six and a two has to be there.
04:40
So there's only two outcomes that do that out of the e probability was 536...