00:01
So, in this question, we're told that we're rolling two fair dice separately, and we're also told that each die has six faces.
00:08
In part a, what we're asked to do is we're asked to list our sample space.
00:12
So, we know that our sample space is just the collection of all possible outcomes we can get, right? so if i had a spinner, and our spinner had, like, three sections, you know, one was orange, one was blue, and one was red, if we were to just have our spinner, and our spinner.
00:33
Spinner we hear and we were to spin it, how many outcomes can we get? and what are those outcomes, right? so it would obviously be, you know, red, orange, or blue, because this is a collection of all the possible outcomes we can get by spinning our purple spinner right here.
00:55
Okay? so the same sort of logic applies here.
00:57
We know that we're flipping, sorry, rolling our dice separately, right? we're supposed to list the sample space or just like list all the possible things we can get, right? so let's go through this logically.
01:11
If we roll a 1 first, what can we roll afterwards? well, we can roll a 1, we can roll a 2, we can roll a 3, we can roll a 4, we can roll a 5, we can roll a 6.
01:37
Okay, well, what if instead of rolling a 1, we rolled a 2? well, if you roll the 2 first, then we could roll a 1 afterwards.
01:44
That's still a possibility.
01:46
We could roll a 2.
01:49
We could roll a 3.
01:54
We could roll a 4.
01:58
We could roll a 5.
02:01
Or we could roll a 6.
02:03
Okay? what we do is that we end up continuing until we get to our final one, which is a 6, right? so we keep on applying this pattern of putting our first number first and then going in order from 1 to 6.
02:30
And that's going to give us all of our sample spaces.
02:32
Okay? so, just give me one second to fill this in.
02:38
Okay, so i'm back and i finished with our sample space.
02:42
We just followed the same pattern that we did in our first two rows.
02:45
So, now that we've listed our sample space, we can move on to part b.
02:51
And in part b, we're told that event a is the event where we roll a 3 or 4 first, and then followed by an even number, right? what we're asked to do here is to determine the probability of event a occurring, okay? so let's look for all the outcomes in which we have a 3 or 4 first, right? so we can start eliminating.
03:14
So we know that in this row right here, so in this row and in this row, we're starting with either a 3 or 4 first.
03:23
Okay? so let's just isolate this.
03:28
Let's start dealing with our part b.
03:31
Whoops.
03:34
I did not mean to do that.
03:41
Let's just put it here for our part.
03:43
And to end up with our event a, let's find all the outcomes in here that are followed by an even number.
03:52
Okay? so we know that over here we follow with an odd number.
03:56
We do that over here and over here.
04:00
And that leaves us with these six outcomes, right? so it means that out of our 36 total possible outcomes, six of them have the trait we want.
04:12
So it means that the probability of us selecting or of us having this event, of this event occurring pretty much, is just 6 over 36, which is about 0 .167 if you want it to round.
04:26
Okay? so that's our part b.
04:29
Now, i'll raise this from up here.
04:34
In part c, we're told that event b is the event where the sum of two roles is at most 7.
04:40
What we're asked to do here is define the probability of event b.
04:43
Okay? so let's keep this in mind.
04:49
Let's copy this and let's start determining our sums.
04:57
So if you're telling us that the sum of our two roles has to be at most seven, then it means that our first outcome, which we can call f, added to our second outcome, which we can call s, has to be less than or equal to seven.
05:16
So we're looking for all numbers here that have an outcome of at most seven.
05:23
So let's start from over here at the bottom.
05:25
6 plus 6, well, that's definitely greater than 7, so it doesn't satisfy our inequality over here, so we can get rid of it.
05:35
Same for 6 plus 5 and 5 plus 6.
05:40
Same for 5 and 5, that ends up being 10.
05:44
6 .4, 4, 6 .6.
05:46
6 .3, well, that's going to be 9.
05:49
5, 4.
05:50
Also 9.
05:52
Also 9.
05:53
Also 9.
05:54
Okay.
05:55
6 .2, that's 8.
05:57
5 .3.
05:57
3 is also 8, 4 4 is 8, 3 5 is 8, and 2, 6 is 8.
06:05
Okay? and now, finally, 6 -1 is 7, 527, 437, all these outcomes here are 7.
06:15
And then we also have our outcomes here, which are obviously going to be smaller than the others.
06:23
We can also, if you want to, you can pause the video and just manually verify that all of these are less than equal to.
06:31
To 7.
06:32
So i'll give you a minute to do that.
06:36
And now that you've determined that, we can see that there are a total of 21 ways where we can roll a sum of at most 7, right? so if there are 21 ways this can happen, and there are 36 ways we can actually roll our dice, then the probability that we have a sum with a sum of two rolls being at most 7, it's just going to be 21 over 36, or about 0 .853, sorry, 0 .583.
07:09
Okay, so that's our part c.
07:13
Finally, well, not finally, in part d, what we're asked to do is we're asked to explain what the probability of a given b represents and to find it.
07:29
So, by definition, we know that we can find our probability of a given b or the probability of event a occurring given that event p, has already occurred by finding the probability of events a and b occurring at the same time, divided by the probability of event b occurring.
07:50
Okay? so we know how to calculate it.
07:52
If you wanted to translate our situations into our words, you know, we know in this case we're looking at the probability of a given b.
08:04
So we can start with that.
08:06
Probability of a given b has occurred, right? so instead of a and b, let's sub in the requirements of our situation.
08:19
We know that in event a, we are looking for the event where a three or four is rolled first...