Quadratic inequality word problem

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 2 and 1/4 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 a row, or really any other candy twice in a row, is greater than 2 and 1/4 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 0 and 1. So we want to write the inequality that models the 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 its 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 if p is the probability of getting "Honey Bunny," what's the probability of getting any other candy at once? Well, that's going to be 1 minus p. If you have a probability of p of getting "Honey Bunny, well, then it's 1 minus p of anything but "Honey Bunny." Now, what's the probability of getting this twice in a row, of 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 1 minus p, 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 other non-"Honey Bunny" candy, any other candy, twice in a row. Now, they tell us that this probability needs to be greater than 2 and 1/4 times the probability of getting "Honey Bunny" in one try. So is greater than 2 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-- well, I'll just multiply it out. So this is going to be 1 squared minus 2p plus p squared. And that's going to be greater than 2 and 1/4 p. Now let's see. If 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 2p minus 2 and 1/4 p, so that's going to get us minus 4 and 1/4 p. Or let me just write that as 17/4 p plus 1 is greater than 0. And so let's think about solving this quadratic right over here. And under which circumstances is this greater than 0? To 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 17/4 right over here, so let's multiply both sides times 4. And since 4 is a positive number, it's not going to change the sign, the direction of this inequality. So we could rewrite this as 4p squared minus 17p plus 4 is greater than 0. 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 of 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-- let's see, 17 minus 15 over 8 is 2/8-- which is equal to 2/8 or 1/4. So that's one of them. That's when we take the minus. And if we add 17 plus 15, that gets us to 32 divided by 8 is 4. So there's two situations right over here. Let's factor this out. We could write this as p minus 1/4 times p minus 4 is greater than 0. So under what circumstances 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 0, 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. And I'll switch colors here just for fun-- so both positive or both negative. So if they are both positive-- let me 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 1/4, and p is greater than 4. So that's the situation where they are both positive. Now, what about if they are 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 4. Now, what does 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 four. And this is an and right over here, so this collapses to p is a 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. To go back to the original problem, we're talking about a probability of someone getting "Honey Bunny" in one try. A probability must be between 0 and 1, so the probability 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 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 the inequality 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 needs to be greater than 2 and 1/4. So we could write that multiple times. I could write that as 2.25 times p. And then the probability of getting "Honey Bunny" in one try must be less than 0.25. And we're done.