### Video Transcript

Monica’s padlock has a four-digit code. Each digit is a number from zero to nine. Her brother has changed the code on her padlock. He may have repeated digits in the code. She knows that the first three digits are nine, eight, seven. Part a) Monica chooses a digit at random. What is the probability that she correctly chooses the final digit of the code?

There is also a part b which we’ll come on to. Well, to help us solve this problem, we need to know a little bit about probability. Well, we know that the probability of an event, and we can mark it like this, 𝑝 and then event in brackets, is equal to the number of successful outcomes divided by the number of possible outcomes. So we can apply this to our problem.

Well, first of all, we’re gonna work out the number of possible outcomes. Well, we know that the possible outcomes are zero to nine. And we know that it can be any of the digits from zero to nine because it says that her brother may have repeated the digits in the code. So having nine, eight, seven does not rule any of them out. So now, we can see that the total number of possible outcomes is 10 because if we’re going from digits zero to nine, this is 10 digits. Be careful, because a common mistake here would be to just say nine. But in fact, it’s 10 because we include the digit zero.

So therefore, we can see that the probability of the correct digit being chosen is gonna be equal to a fraction with 10 as the dominator. And that’s because that’s the number of possible outcomes. But what we now need to know is what is the number of successful outcomes.

Well, for the padlock code to be correct, it needs to be the exact four digits for the code is. So therefore, there is only one digit that’s going to be correct. So we can say that the number of successful outcomes is gonna be one. So therefore, the probability that the correct digit is chosen is one out of 10 or one-tenth. So for part b, we’re told a little bit more information.

Her brother tells her that the last digit is not a prime number. Monica chooses a number at random which is not prime. What is the probability that she correctly chooses the final digit of the code now?

So we’re told that the last digit is not a prime number. So let’s have a look at which one of the numbers between zero and nine are prime. Before we can point out which ones are the prime numbers, we need to remind ourself what a prime number is. And a prime number is a number that can be divided exactly by itself and one. But it must have exactly two factors. We add that on the end because, without that bit of extra definition, we could say that one was a prime number because it can be divided by itself and one. But one is not a prime number as it doesn’t have two factors.

And therefore, our prime numbers are two, three, five, and seven cause we know that zero is not cause it cannot be divided by one. One we’ve already said isn’t a prime number. Four isn’t because it can be divided by two as well as one and four. Six isn’t because two and three are factors. Eight isn’t because two and four are factors. And nine isn’t because three is also a factor. So therefore, the numbers that she can choose are zero, one, four, six, eight, and nine as these are not prime numbers.

So therefore, we can use this to work out the probability that she correctly chooses the final digit of the code now because, as we said before, the probability of an event is equal to the number of successful outcomes over the number of possible outcomes. So therefore, the probability that the correct digit is chosen is gonna be equal to one-sixth or one over six. And that’s because the number of successful outcomes is just one because there can only be one correct digit. But this time, there are six possible digits to be chosen from. And that’s because, as we said, there’s zero, one, four, six, eight, and nine now that she can choose from.

So as we’ve said, the correct probability is one-sixth.