An integer from one to 100 is chosen at random. Find the probability that it is part i) divisible by eight and part ii) not divisible by eight.
We begin this question by recognizing that our sample space consists of the integers from one to 100. The elements in our sample space, which we’ll call 𝑆, are therefore one, two, three, and so forth, all the way up to 100. And it should be easy to see that the number of elements in 𝑆 is 100.
Let us now define an event which we’ll call 𝐴. We will say that 𝐴 is the event that the integer that we choose from our sample space is divisible by eight. To find the elements of set 𝐴, we now ask the question which of the integers from one to 100 will be divisible by eight.
We now recall that, in order for a number to be divisible by eight with no remainder, by definition, it is part of the eight times table. We can then find the elements of set 𝐴 simply by writing out the eight times table. Eight times one is eight, eight times two is 16, eight times three is 24, and so on, all the way to eight times 12, which is 96.
If we were to continue, we would get eight times 13, which is 104. Since 104 is greater than 100, it is not part of our sample space. So we get rid of this and stop at 96. We can now see that since we went from eight times one to eight times 12, we therefore have 12 elements in the set of 𝐴.
And now on to probabilities, the question tells us that the integer from one to 100 is chosen at random. This means that each of the integers has an equal probability of being chosen. Because of this, we can find the probability of 𝐴 by dividing the number of elements of 𝐴 by the number of elements in our sample space.
Subbing in the values that we have found, we find the probability of 𝐴 is therefore 12 over 100. And dividing the top and bottom half of this fraction by four, we get that the probability of 𝐴 is equal to three over 25. We therefore have our answer to part i of the question. The probability that the integer chosen is divisible by eight is three over 25.
Let’s put this value to one side and now move on to part ii of the question. Find the probability that the integer chosen is not divisible by eight.
Now we’ve already defined that if event 𝐴 happens, the number chosen is divisible by eight. And therefore, if event 𝐴 does not happen, then our number will not be divisible by eight.
part b [part ii] of this question, we can therefore find the probability of the complement to event 𝐴, 𝐴 dash. Since the probabilities of all possible events sums to one, we know that the probability of 𝐴 dash is equal to one minus the probability of 𝐴. We’ve already found that the probability of event 𝐴 is three over 25. And we can sub this value in. A one is the same as 25 over 25. And therefore, our solution is 22 over 25. We have therefore answered part ii of the question. And we found that the probability that the integer chosen is not divisible by eight is 22 over 25.