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## Statistics and probability

### Unit 7: Lesson 7

Multiplication rule for independent events- Sample spaces for compound events
- Sample spaces for compound events
- Compound probability of independent events
- Probability of a compound event
- "At least one" probability with coin flipping
- Free-throw probability
- Three-pointer vs free-throw probability
- Probability without equally likely events
- Independent events example: test taking
- Die rolling probability with independent events
- Independent probability
- Probabilities of compound events
- Probabilities involving "at least one" success
- Probability of "at least one" success

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# Die rolling probability with independent events

We hope you're not a gambler, but if you had to bet on whether you can roll even numbers three times in a row, you might want to figure this probability first. Created by Sal Khan and Monterey Institute for Technology and Education.

## Video transcript

Find the probability of rolling
even numbers three times using a six-sided die
numbered from 1 to 6. So let's just figure out the
probability of rolling it each of the times. So the probability of rolling
even numbers. So even roll on six-sided die. So let's think about
that probability. Well, how many total
outcomes are there? How many possible rolls
could we get? Well, you get one, two, three,
four, five, six. And how many of them satisfy
these conditions, that it's an even number? Well, it could be a 2,
it could be a 4, or it could be a 6. So the probability is the events
that match what you need, your condition for right
here, so three of the possible events are an even roll. And it's out of a total of
six possible events. So there is a-- 3 over 6 is
the same thing as 1/2 probability of rolling
even on each roll. Now they're going to
roll-- they want to roll even three times. And these are all going to
be independent events. Every time you roll, it's not
going to affect what happens in the next roll, despite what
some gamblers might think. It has no impact on what happens
on the next roll. So the probability of rolling
even three times is equal to the probability of an even roll
one time, or even roll on six-sided die-- this thing over
here is equal to that thing times that thing again. All right, that's our first
roll-- we copy and we paste it-- times that thing and then
times that thing again. Right? That's our first roll,
which is that. That's our second roll. That's our third roll. They're independent events. So this is going to be equal to
1/2-- that's the same 1/2 right there-- times 1/2
times 1/2, which is equal to 1 over 8. There's a 1 in 8 possibility
that you roll even numbers on all three rolls. On this roll, this roll,
and that roll.