00:01
We can use this sample space to think about theoretical probabilities, such as what's the probability of rolling a sum of seven? so you can see there's six possible ways to do that.
00:10
So the probability of getting a sum of seven.
00:14
Out of the total 36 possible outcomes, six of those gave us a sum of seven.
00:20
So one out of six gives us about a .160 chance.
00:25
If we were to carry out an experiment, say take two dice and roll them ten times.
00:30
Or 50 times or 100 times, there's no guarantee that those experimental probabilities will line up exactly with our theoretical probability, especially in the short term.
00:41
If we only roll the dice a few times, we'll probably be pretty far off from the theoretical probability, but as we roll more and more, we do more trials, we will approach that theoretical probability.
00:52
This is called the law of large numbers.
00:55
If we're recording the total sum, then we have a probability distribution, something like this, but if we specifically wanted to know what's the probability of getting a sum of 7, we could set this up as a binomial random variable.
01:09
To check and see if a binomial distribution is appropriate, we can check the bins.
01:14
We need a situation that is binary.
01:16
So we either want a sum of 7 or not.
01:21
So we do have a binary situation.
01:23
We need independence, which we do.
01:25
Every time we roll the die, it does not affect the next roll.
01:31
We need to know the number of trials, which we will, and we need the same chance of success each time.
01:37
So we should be able to set this up as a binomial distribution where we're wondering, did we get a sum of seven? so we found that there's a one out of six chance of that happening, which means the complement would be five out of six.
01:54
There's our probability table.
02:01
So we're going to let x be a binomial random variable representing this situation.
02:05
So we can answer questions like, what's the probability that we would get four sevens out of ten rolls? out of ten trials, we want four successes or four sevens.
02:18
The probability of success is one out of six, and we want four of those.
02:22
And the probability of failure is five out of six, and we want six of those.
02:30
You can find that formula on an ap formula chart here, where n represents the total number of trials.
02:37
P is the probability of success, and x is the number of desired successes.
02:44
This 10 over 4 notation means the combination.
02:47
So we would do 10 factorial divided by 4 factorial times 10 minus 4 factorial.
02:56
And we'll use a calculator to help us out with that.
02:59
So if we throw this all into a calculator, we get a probability.
03:02
The chance of having 4 -7s out of 10 would be 0 .054...