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CCSS.Math:

lisa owns a random candy vending machine which is a machine that picks a candy out of an assortment in a random fashion lisa controls the probability in which each candy is picked she is running out of honey bunny so she wants to program its probability so that the probability of getting a different candy twice in a row is greater than two and one-fourth times the probability of getting honey bunny in one try so let me read that again she wants to program its probability so that the probability of getting a different candy twice in her or really any other candy twice in a row is greater than two and one-fourth times the probability of getting honey bunny in one try write an inequality that models the situation use p to represent the probability of getting honey bunny in one try solve the inequality and complete the sentence remember that the probability must be a number between zero and one so we want to write the inequality that models a problem here and then we want to complete the sentence the probability of getting honey bunny in one try must be so they give us a bunch of options greater than greater than or equal to less than less than or equal to and then we have to put some number here so to work through this I've copy and pasted this problem onto my little scratch pad right over here and so let's just think about it a little bit so they tell us use P to represent the probability of getting honey bunny in one try and they also say she wants to program his probability so that the probability of getting a different candy twice in a row is greater than 2 and 1/4 the probability of getting honey bunny in one try so what's if P is the probability of getting honey bunny P is the probability of getting honey bunny what's the probability of getting any other any other candy at once well that's going to be one minus P if you have a probability of P of getting honey bunny well that is 1 minus P of anything but honey bunny now what's the probability of getting this twice in a row getting anything else twice in a row well you're just going to multiply this probability times itself it's going to be 1 minus P times - pee or we could just write that as 1 minus P squared so this right over here is the probability of getting a different candy any other candy twice in a row so prob of any any other non honey-bunny candy other any other candy twice in in a row now they tell us that this needs to be greater than this probability needs to be greater than two and one-fourth times the probability - and 1/4 so probability of getting honey bunny in one time in one try so 2 is greater than 2 and 1/4 - and 1/4 times the probability of getting honey bunny in one try well that is P times P so we have just set up the first part we have written an inequality that models the situation now let's actually solve this inequality and so to do that I will just expand 1 minus P squared out 1 minus P squared is the same thing as P squared is the same thing as is the same well I'll just multiply it out so this is going to be 1 squared 1 squared minus 2p plus P squared and that's going to be greater than 2 and 1/4 P greater than 2 and 1/4 P now let's see if we subtract we subtract 2 and 1/4 P from both sides we're going to be left with and I'm going to reorder this we're going to get P squared so you have minus 2 P minus 2 and 1/4 P so that's going to give us minus 4 and 1/4 P R let me just write that as 17 over 4 P plus 1 plus 1 is greater than greater than 0 is greater than 0 and so let's think about let's think about solving this quadratic right over here and under which circumstances is this greater than 0 think about it let's factor it and actually before we factor it let's simplify it a little bit I don't like having this seventeen fourth right over here so let's multiply both sides times four and since 4 is a positive number it's not going to change the sign of the direction of this inequality so we could rewrite this as 4p squared minus 17 P plus four plus four is is greater than zero is greater than zero and let's see what are the roots of this and we could use the quadratic formula if we wanted to do it really quick we could probably do it other ways but negative B so it's going to be 17 plus or minus the square root the square root of of 17 negative 17 squared B squared so that's 289 minus 4 times a times C well a times C is 16 times 4 so minus 64 all of that over 2 times a all of that over 8 so that's 17 plus or minus let's see this is the square root of 225 over 8 which is equal to 17 plus or minus 15 over 8 which is equal to well let's see 17 minus 15 over 8 is 2 8 which is equal to two eighths or 1/4 so that's one of them that's that when we take the minus and if we add 17 plus 15 that gets us to 32 divided by 8 is 4 or 4 so there's two situations right over here so we could let's let's factor this out we could write this as P minus 1/4 times P minus 4 is greater than is greater than 0 so under what circumstances is is this going to be true what constraints are this going to be true well if you're taking the product of two terms and they are going to be greater than zero that means that these two things have to be the same sign or actually in particular they both have to be positive or they both have to be negative so let's look at those two situations so I'll switch colors here just for fun so both positive or both both negative so if they're both positive that means that P we do it here that means that P minus 1/4 has to be greater than 0 and P minus 4 is greater than 0 add 1/4 on both sides right over here you get P is greater than is greater than 1/4 and P P is greater than 4 so that's the situation where they are both positive now what about if they're both negative well then you have P minus 1/4 is less than 0 and P minus 4 is less than 0 add 1/4 here so P needs to be less than 1/4 and P needs to be less than it needs to be less than 4 now what do what does this expression what is this constraint simplify to P has to be greater than 1/4 and P has to be greater than 4 well if P is greater than 4 it's definitely going to be greater than 1/4 so all of this collapses into P needs to be greater than 4 that's the situation where both are positive P must be greater than 4 now what about here well if P is less than 1/4 it's definitely going to be less than 4 and this is an end right over here so this collapse is 2 P is less than P is less than 1/4 so which one do we go with P needs to be greater than 4 or P needs to be less than 1/4 well we need to remind ourselves that we're talking about a probability back to the original problem we're talking about a probability of someone getting honey bunny in one try probability must be between 0 and 1 so the probability being having to be greater than 4 well that just doesn't make any sense that doesn't make any sense in the context of this question so we have to go with the probability of getting honey and honey bunny needs to be less than 1/4 or less than 0.25 which makes complete sense so let's fill in this information this was this was the inequality that modeled that modeled the problem and we got P has to be less than 1/4 so let's go back to the original problem the inequality was 1 minus P squared 1 minus P squared needs to be greater than 2 and 1/4 so we could write that multiple times I could write that as 2 point 2 5 times P and then the probability of getting honey bunny in one try must be less than less than 0.25 and we're done