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# Die rolling probability

## Video transcript

We were given two dice to roll. One is black with six sides. The other is white
with four sides. So a four-sided die is kind
of a pyramid-looking thing. It has exactly four sides. It would be a pyramid
with a triangular base. For a given roll, what
is the probability that the dice add up to 10? If it helps, you may select
the matching outcomes below. Your selections aren't
checked with your answer. So this might look a little
bit strange at first. You're like, what
is this triangle? This triangle is the face
of the four-sided die that is facing you. So this is the roll
of the four-sided die. So this is a roll of 1 and 1. So these clearly are
not adding up to 10. So let's think about all
the ones that add up to 10. So this adds up to 2. This adds up to 3. This adds up to 4. This adds up to 5. Let's see what
actually adds up to 10. So if we go all the way-- so
clearly, if we get a 4 and a 6, that's going to add up to 10. And are there any
other possibilities that add up to 10? Well, the highest possible roll
I can get on a four-sided die is a 4. So that's my best scenario. That's the highest
possible roll. And even if I get that, I still
need the highest possible roll on the six-sided die. So this is the only
scenario that adds up to 10. You can look at any
of these other ones. They'll add up to
something less than 10. So there's one possibility
that satisfies our constraints. The dice add up to 10. And how many total equally
likely possibilities were there? Let's see. This grid shows all the
equally likely possibilities. This is 1, 2, 3, 4
times 1, 2, 3, 4, 5, 6. There are 24 equally
likely possibilities, and that comes from you
have 6 possibilities from the six-sided die
times 4 possibilities from the four-sided
die gives you 24 equally likely possibilities. Only one satisfies
the constraint that the dice add up to 10. Let's do a few more of these. So you're given two
six-sided dice to roll. For a given roll, what
is the probability that the dice add to 6? So here we have the
tops of both of my dice. And now what are all the
scenarios where the dice add up to 6? Well, let's think about this. If I have a 1 and a 5,
that's going to add up to 6. In this entire column,
that's the only way that I add up to 6. Right over here, let's see. If I have a 2 on the black
die, then I'm going to need a 4 on the white die to add up to 6. If I have a 3 on the black
die, I'm going to need a 3 on the white die to add up to 6. If I have a 4 on the black
die, I'm going to need a 2 on the white die. If I have a 5 on the black
die, I'm going to need a 1 on the white die. And then if I have 6
six on the black die, it's actually impossible that
the sum will be exactly 6. So there's no zero
on the white die. So none of these
meet the constraint. So here I have one, two,
three, four, five possibilities where my dice add up to 6. So I have five
possibilities out of a total of 36 equally likely ones. This is a six by
six grid, and it comes from six
possibilities for one die times six possibilities
for the other die. So 5 possibilities
satisfy my constraint out of a total of 36. Let's do one more. This is actually a lot of fun. You're given two
six-sided dice to roll. For a given roll, what
is the probability at least one die is a six? So let's see. So we just have to look
for all of the situations where at least one
of the dice is a six. So that one, that one. Every time I see a six, I
should just click on that. That one. That one. And then obviously all of these
scenarios right over here. So these are all of the
scenarios where at least one of the dice is a six. Here, of course, is where
both of the dice are a six. So how many scenarios are there? 1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11. So there are 11
possibilities out of a total of 36
equally likely ones. Remember, 6 times 6. There's 36 equally
likely outcomes. So let's check our answer. And we got it right.