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### Course: Statistics and probability > Unit 7

Lesson 2: Probability using sample spaces# Subsets of sample spaces

Sal solves an example about subsets.

## Want to join the conversation?

- At4:30, these questions should be reworded as "the subset consists of ONLY outcomes where...". to represent Sal's logic, otherwise this is not grammatically correct. The sample subset DOES contain ALL the outcomes where your friend wins OR there is a tie. It's extremely confusing and it is used in the quiz after the video as well.(35 votes)
- My thoughts are the same as yours. I believe that what Sal explains is not correct. The first 3 statements are actually to tell us to check what's not included in the subset to see whether or not they contain what meets the question. And if it does, then the statement is false - because the subset doesn't contain all stated outcomes. Based on this logic, the correct answer is No. 4 still.(15 votes)

- Question: is there a practice task called "Describing subsets of sample spaces"? I was told to do this by my teacher but I can't find it; I can only find this video. Also, it was recommended for me but does not load. I try to do it and it sends me back to the homepage. Thanks!(10 votes)
- Around4:30, he talks about option one, "The subset consists of all the outcomes where your friend does not win."

To me, that does not require all the outcomes selected to be ones in which your friend does not win, but rather that the subset contains all the possibilities in which your friend does not win. It could include all the possibilities in which your friend does not win, plus some. His reasoning for not selecting that option was that the subset includes an outcome in which your friend does win. This seems faulty to me.

That is how I read that option on a purely grammatical level. Is there some mathy jargon I am missing here?(10 votes)- Yes. Technically, if the friend wins, it isn't faulty. He won.(3 votes)

- also, does it matter in what order you write the different outcomes?(6 votes)
- I don't think so. It just simplifies (or not) your way to solve the problem.(6 votes)

- is there also an equation for these kind of problems without making a chart?(0 votes)
- Well, you CAN write it formally without using a chart, but you would still need to solve the probabilities in your head.

For the Harry Potter example, you would write the first option as P (Holly ∪ Unicorn) - That U in the middle means you're calculating a total probability (it really means "union", as you've probably seen when dealing with inequalities in algebra). That can be solved as P(Holly) + P(Unicorn) - P(Holly and Unicorn happening together). This is where you need to make your chart in your mind: How many times can a Holly wand happen? How many times can a Unicorn wand occur? And lastly, how many times can a Holly wand and a Unicorn wand happen together at the same time? You know a Holly wand can happen 5 times out of 20, so P(Holly) = 5/20. A Unicorn wand can happen 4 times out of 20, so P(Unicorn)= 4/20. And a Holly AND Unicorn wand can only happen one time out of 20, so P(Holly AND Unicorn) = 1/20.

Getting back to our original equation, P(Holly) + P(Unicorn) - P(Holly and Unicorn happening together) = 5/20 + 4/20 - 1/20 = 8/20. And that's the same result you get when considering the chart.

P.S.: You're subtracting the P(Holly and Unicorn) because you're counting that outcome twice, when considering the single probabilities of P(H) and P(U).(16 votes)

- It says: "The wand that selects Harry will be made from of holly or unicorn hair."

Wouldn't this exclude the wand that is both? I mean, shouldn't "holly OR unicorn" mean that it can't be both? (It is either holly or unicorn)(4 votes)- In mathematics, "or" is an inclusive or. So "A or B" would mean "A or both or both".(3 votes)

- I am still a little confused. What does subsetes even mean?(3 votes)
- In this case, the word "set" means the set of all possible outcomes. in the case of flipping a coin, the set of possible outcomes is either heads or tails. A subset is a smaller set of outcomes that is contained in that larger set. So if you are rolling a die and what to know the probability of getting an odd number, you are looking at the subset that contains all the outcomes in which the die comes up with an odd number (1, 3, and 5). Hope that helps!(4 votes)

- A quick question:

Where did you get those questions.Please give a link

Thank you!

Simon(1 vote) - I really don't understand in the first example why the or is 8/20 and not 9/20. It said Holly OR Unicorn hair. The chances of picking Holly would be 5/20, and the chances for unicorn hair would be 4/20. If you add those you get 9/20 and not 8/20. So what's the deal? I know he refused to count the Holly and Unicorn twice but I don't get why. It is like you are only counting the Unicorn 3 times then...(1 vote)
- There are 5 wands that use Holly. That's 5/20 wands. Then there are 4 wand's that use Unicorn Hair. However, one of the wands that uses Unicorn Hair also uses Holly. Since there is only one wand with Holly and Unicorn Hair and its already been counted (when we counted the ones with Holly), we can't count it again. Therefore, we have 3/20 wands. Adding that to our 5/20 wands, we get 8/20 = 2/5. You probably already knew that. I just wanted to be sure we were on the same page.

Now, think about if there were only one wand, and it was made with Holly and Unicorn. That's 1/1 wands with Holly and 1/1 wands with Unicorn. 1/1 + 1/1 = 2 wands, but that doesn't mean that there are two wands. We simply have counted one of the wands twice. That is what Sal is doing when he skips that wand the second time around. Makes more sense?(3 votes)

- At5:10, it gets a little confusing. I was wondering why Sal was going over the outcomes of a tie for the last option, and why those were ruled out.(1 vote)
- because we went through the tie outcome in the second line(1 vote)

## Video transcript

- So, this right over
here is a screenshot of the Describing Subsets
of Sample Spaces exercise on Khan Academy, and
I thought I would do a couple of examples, just
because it's good practice just thinking about how do
we describe sets and subsets. So it reads, Harry Potter
is at Ollivanders Wand Shop. As we all know, the wand
must choose the wizard, so Harry cannot make the choice himself. He interprets the wand
selection as a random process so he can compare the probabilities
of different outcomes. The wood types available are holly, elm, maple, and wenge. The core materials on offer are phoenix feather, unicorn
hair, dragon scale, raven feather and thestral tail. All right! Based on the sample space of
possible outcomes listed below, what is more likely? And so, we see here, we have
four different types of woods for the wand, and then each of those could be combined with five
different types of core, Phoenix Feather, Unicorn
Hair, Dragon Scale, Raven Feather and Thestral Tail. And so, that gives us four different woods and each of those can be combined
with five different cores. 20 possible outcomes. And they don't say it here but they way they're talking I guess we can, I'm going to go with the assumption that they're equally likely outcomes, although it would have been
nice if they said that, "These are all equally likely" but these are the 20 outcomes. And so, which of these are more likely? The wand that selects Harry will be made of holly or unicorn hair. So, how many of those
outcomes involve this? So if, Holly are these five outcomes and if you said, "Holly or Unicorn Hair" it's going to be these five outcomes plus, well this one involves Unicorn Hair but we've already included this one, but the other ones that's
not included for the Holly, that involve Unicorn
Hair, are the Elm Unicorn, the Maple Unicorn and the Wenge Unicorn. So it's these five, plus
these three, right over here. So eight of these 20 outcomes. And if these are all
equally likely outcomes, that means there is an
8/20 probability of a wand that will be made of
Holly or Unicorn Hair. So this is 8/20 or, that's the same thing as 4/10 or 40% chance. Now, the wand that selects Harry will be made of Holly and Unicorn Hair. Well, Holly and Unicorn Hair, that's only one out of the 20 outcomes. So this, or course, is going
to be a higher probability. It actually includes this outcome and then seven other outcomes. So, the first choice includes the outcome for the second choice
plus seven other outcomes. So this is definitely going
to be a higher probability. Let's do a couple more of these or at least one more of these. You and a friend are
playing "Fire-Water-Sponge". I've never played that game. In this game, each of the two players chooses fire, water or sponge. Both players reveal their
choice at the same time and the winner is determined
based on the choices. I guess this is like
Rock, Paper, Scissors. Fire beats sponge by burning it. Sponge beats water by soaking it up. And water beats fire by putting it out. Alright, well, it kind of makes sense. If both players choose the
same object, it is a tie. All the possible outcomes of
the game are listed below. If we take outcomes
one, three, four, five, seven and eight, as a
subset of the sample space, which of the statements
below describe the subset? So let's look at the outcomes
that they have over here. Well, it makes sense that there are nine possible outcomes because, for each of the three choices I can make there's going to be three
choices that my friend can make. So, three times three is nine. They've highlighted these red outcomes. Outcome one, three, four,
five, seven and eight. So let's see what's common about them. Outcome one, Fire, I get
Fire, friend get's Water. OK, so let's see, my friend would win. Outcome three, I pick Fire,
my friend does sponge, so actually I would win that one. And then outcome four, Water, Fire. And then outcome five, Water, Sponge. I don't see a pattern just yet,. Let's see what the choice is. The subset consists of all outcomes where your friend does not win. All outcomes where your
friend does not win. Well, that's not true because look, outcome one my friend wins. Water puts out fire so, we're not going to
select this first choice. So let's see, the subset consists of all the outcomes where your friend wins or there is a tie. So let's see. Where the friend wins or there's a tie. Well, outcome three, this is
an outcome where I would win. Or you, or whoever "Your" is, whoever they're talking about. This is one where the friend doesn't win because Fire burns Sponge so, I'm not going to select that one either. Choice three, the subset consists of all of the outcomes where
you win or there is a tie. Well, we just said, "Outcome one, I don't win that, my friend wins that. Water puts out the Fire." Now let's look at the last choice. The subset consists of all the outcomes where there is not a tie. Alright, so this is
interesting because look, outcome two there is a tie. Outcome six, there is a tie. Outcome nine, there is a tie. There's actually only three
scenarios where there's a tie. Either it's Fire, Fire. Water, Water. Or Sponge, Sponge. And those are the ones
that are not selected. So, all of these, someone is going to win. Outcome one, three, four,
five, seven or eight. So, definitely, definitely
go with that one.