Subsets of sample spaces

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 winged whinge the core materials on offer are phoenix feather unicorn hair dragon scale raven feather and thestral tail alright 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 for five different cores 20 possible outcomes and they don't say it here but the way they're talking I guess we can I'm gonna go with the assumption that they're equally likely outcomes although it would have been nice if they said that these are all equally equally likely but these are the 20 outcomes and so which of these are more likely the one 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 gonna be these five outcomes plus well this one involves unicorn hair but we've already we've already included this one but the other ones that's not included for the holly that involved unicorn here are the elm unicorn the maple unicorn and the winds 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's an eight twenty it's probability of a wand that it will be made of Holly or unicorn hair so this is eight twenty eight so that's the same thing as four 10s or 40% chance now the one 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 of course is going to be a higher probability it actually includes this outcome and then seven other outcomes so this is this the first choice is this includes the outcome for the second choice plus seven other outcomes so this is definitely going to be a higher probability let's do let's do a couple more of these or at least one of them one more of these you and a friend are playing fire water sponge and ever 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 kind of make sense if both players choose the same object that is a tie all the possible outcomes of the game are listed below if we take outcomes 1 3 4 5 7 & 8 as a subset of the sample space which of the following statements which of the statements below describe this 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 the friend my friend can make so three times three is nine let's see they've outcome they've highlighted these red outcomes outcome 1 3 4 5 7 & 8 so let's see what's common about them outcome 1 fire I get fire friend gets water okay so let's see my friend would win outcome 3 I pick fire my friend does sponge so actually I would I would win that one and then outcome for water fire and then outcome 5 water sponge sponge huh these are all I don't see a pattern just yet let's see the choices 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 1 come one my friend wins water puts out fire so that's we're not gonna select this first choice so let's see the subset consists of all the outcomes where your friend wins or there's a tie so let's see where the friend wins or there's a type of outcome three this is this is one where this is an outcome where I would win where are you or whoever your is where they're talking about this is one where the friend doesn't win because fire burns sponge so I'm not gonna select that one either choice three the subset consists of all of the outcome 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 puts out the fire now it's the glass 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 spun 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