# Product Rule and Independent Events

## Transcript

Single digits are being generated randomly, so like these kind of numbers. It doesn’t have to be a five-digit number but these are the kind of things randomly generated. It can be any random number, okay? And it can be repeated or there’s no limit to this. So it’s like these kind of numbers. These are just examples. In part a! I want to find the probability that the first three digits are one.

Well, one! How many numbers can we choose from by the way? We can start from one two three four five six seven at nine and zero. There’s ten numbers we can choose from. And I want the first three digits to be one. We want the probability of having one for the first number and one for the second number and one again for the third number. And we know that the probability of getting one is just 1 on 10, isn’t it? It’s simply 1 on 10 times 1 on 10 times 1 on 10. It’s going to be same for the first second and third numbers because one can be repeated. There’s no limit to how many times I can use one. So it’s just that. Which is simply that. Okay? And you can see I multiplied because it’s and this and this and this. So multiplying.

b! When the first two digits are less than five. Well, less than five is including one two three, and four. So including zero sorry. So there’s going to be one two three four five possibilities we can have. So if I want the first two digits to be anything out of this anything out of these numbers, it’s going to be probability of having less than 5 for the first number and less than 5 for the second number, so now that less than 5 is going to be 5 out of 10 and having another second number less than 5 is also 5 out of 10 and because it’s this and this, we multiply and you should get 1 on 4 if you simplify it, okay? That’s it. That’s pretty much it. So again just applying our multiplication rule.

Now c! If the first three digits are 3, and the probability or we want to find the probability that the next digit is a 3. So they said they give us information that the first three digits numbers were already 3. But we want to find the probability that the next digit is a 3 as well. But guys, before you start think to yourself. Does it matter what the previous numbers were? Does the next number depend on what the previous numbers were? No! All the numbers in all the digits are independent of each other. So even if the previous number was three, even if the previous number was zero, even if the previous number was nine, it doesn’t matter. The next number will always be independent. The probability of getting a three for the fourth number is also as I said it’s independent. So there’s also just 1 on 10. Okay? So because they’re independent. It’s again just 1 on 10. It’s not going to change, okay? So this is just there to trick you, okay? So don’t get tricked by that.

d! If the first four digits are all 9’s, find the problem the next three are 4’s.
So again guys, does it matter if the first four digits were nine or two or zero or two or five? It doesn’t matter! So again they’re just putting that in to trick you guys, so don’t get tricked. So again it’s all independent of each other, so we know that getting a four is 1 out of 10. So if I want to find the probability, I simply multiply 1 out of 10, 1 out of 10, 1 out of 10 for the next three numbers. So you multiply three times which will get you 1 over 1000. A machine can detects car registration numbers with probability of recording error of q. Error of q.

We don’t know what that q is. There’s a probability. Find an expression for the probability that there are no errors when he records n registration numbers. Getting an error is probability of q. The probability of not getting an error is going to be 1 minus q because 1 is the total probability and if I subtract q from that. That will be the probability of having no errors. Because just remember that you can one is a total probability. So if I want to find an expression for the probability that there are no errors for n numbers then it’s like this. no error and no error and no error and no error for n times.

So because AND, we multiply. So it’s going to be 1 minus q times 1 minus q times 1 minus q…. n times, so it’s going to be 1 minus q to the power of n. That’s why we have this expression, so this is your answer, guys. Okay? So have a look guys because it’s AND! Just multiplying that’s why we have a power of n. 