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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 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.