00:01
Okay, young statisticians, i've got a probability experiment with a tree diagram for you today.
00:06
So i have this box of light bulbs, there's 15 of them.
00:10
And inside this box, there are four 40 -watt bulbs, 5 -60 -watt bulbs, and 6 -75 -watt bulbs.
00:18
And i'm going to use the letter f, s, and v for those selections.
00:22
I'm going to pick three out of this box.
00:24
Imagine blindfolded, pick three at random.
00:26
And as i pick them, of course, i'm not replacing them.
00:28
So it's without replacement.
00:30
So if i mapped out in a tree diagram, all possibilities of picking bulbs, the bulb number one, bulb number two, bulb number three, i could get any one of three kinds each time.
00:47
So since i could get any one of three kinds each time, there's 27 outcomes on my probability tree right here.
00:56
So this is not that easy to draw.
00:58
In fact, i've got it quite squished right here.
01:01
But i could fill out this probability tree mapping out all possibilities from getting an f, then an f, then an f, an f, then an f, an f, then an s, an f, then a v, and f, then a v, all the way down through 27 possibilities to v, v, v.
01:20
And then if i were to take my time to map out all these probabilities, i could put them on top of the branches here.
01:26
The chance that i reach in blindfolded and grab a 40 -watt bulb is 4 and 15.
01:31
And now once i set that aside, i have to adjust this fraction because now a 40 watt is set aside, so there's only 14 bulbs left in the box, and only three of them are still the 40 -watt bulbs.
01:43
And now i've set two bulbs aside, so there are 13 left, and only two of them are 40 watts.
01:49
So if i ask you the probability of getting three 40 -watt bulbs, it would be the fractions 4 .15th times 3 .14th times 2 .13th multiplied together.
02:00
Okay, well let's see what probability questions we do need to answer.
02:05
So part a, i want the outcome that exactly two of the bulbs i grabbed are 75 watts.
02:11
So i'm going to take my highlighter here and show you two of them are 75 watts.
02:18
Actually, i will just take my black pen right here.
02:21
That would be here.
02:25
The v is for the letter 75.
02:28
So it would be this one where i'd have an fvv.
02:33
It would be this one where i'd have an svv.
02:40
It would be this one where i would have a v -f -v.
02:48
Whoops, v -f -v.
02:51
It would be this one where i'd have a v -s -v.
02:56
And it would be these two where i'd have v -v -f -f and b -v -v -v.
03:01
So you can see here that i've mapped out six places on the tree that satisfy having two of the 75 watt bulbs and one of another kind.
03:18
So what i'd have to do is write these probabilities down, multiply them out, and add them up.
03:26
So i'm going to pause while i do that.
03:40
Okay, so here is the probability mapped out from the tree.
03:44
Like i said, i identified six different outcomes that matched up with the idea that at least two of the bulbs, sorry, exactly two of the bulbs are 75 watts, which means exactly two or 75s.
03:57
I'm using the letter v for that, and i've got one other bulb.
04:01
Alternatively in red, if you've learned about combinations and permutations or the topic of combinatorics, you can do the idea that from the six bulbs that are the 75 watts, you choose two, that's this notation, six choose two.
04:17
From the nine other bulbs in the box, you choose one.
04:22
And then the denominator is all possibilities, which is 15 total bulbs in the box, choose three.
04:29
So if you work those out individually, you end up with 15 times 9 over 455, which reduces to 27 out of 91.
04:38
Same answer.
04:40
Reducing the fraction after adding those six separate fractions together.
04:45
All right.
04:46
Let's move on to all three being the same rating.
04:49
So if all three of the same ratings, we're talking about getting an fff, or a sss or a vv.
05:06
So remember, for our box, let's look at this picture i drew.
05:12
Here.
05:13
For fff, we've got four of those.
05:16
So it would be a four and fifteen chance times a three out of 14 times a two out of 13.
05:24
For sss we had five of those, so it would be five out of 15 times four out of 14 times three out of 13.
05:36
And for vvv, we had six of those, six out of 15, times 5 out of 14 times 3 out of 13.
05:46
So we multiply and add these fractions up.
05:50
So these denominators are all 2730.
05:58
And if i had the numerators, 24, 60, 90, that's 174 out of 2730.
06:12
All right, feel free to reduce that fraction or change to a decimal.
06:16
And if you wanted to see the alternative method using combinatorics, the fff fraction would look like this.
06:29
From the four bulbs that are f, we choose all three, and from the 11 other bulbs we choose none.
06:39
And then the denominator is 15c3.
06:42
For the sss, from the five bulbs that are 60 watts, we choose three.
06:50
From the 10 other bulbs, we choose none.
06:55
We're 15c3.
06:58
And then for the vvv, from the six bulbs that are v, we choose all three.
07:03
And the nine bulbs, other bulbs, we choose none.
07:08
And again, if you did the alternative way or from the tree diagram method, we're still going to get the same answer.
07:18
And then for the one of each type, the one of each type, if we map it out off the tree, we'd have the outcome.
07:30
Let's see, an f followed by an s followed by a v, or an f followed by a v followed by an s.
07:42
Or we start out with an s, we follow it with an f, then a v.
07:48
An s, a v, then an f.
07:51
Or we start out with a v and get the fs or the v with the sf.
08:04
Sorry that's small and i'm gonna pause to work on the math for this.
08:21
Okay so for the one of each type what you'll notice as you start writing these fractions out they all just worked out to be rearrangements of four times five times six over 15 times so essentially it worked out to 120 at a 2730 times times six, which reduces down to 24 to 91.
08:44
Now the alternative method in red, again, if you've learned combinatorics, this is probably easier...