Two spinners numbered from one to four are spun and the resulting numbers are added together. Part a) Complete the table. Part b) What is the probability of getting a score that is a square number?
All the possible outcomes when these two spinners are spun are represented in the table, which is sometimes called a sample space diagram. The possible outcomes for the number on the first spinner are in the top row of this table. And the possible outcomes for the number on the second spinner are in the first column although, as the two spinners are identical, these could be the other way around.
The numbers inside the table give the result when the scores from the two spinners are added together. We notice first of all that there are two cells on the outside of this table which are blank. The cell marked with an orange star represents the fourth outcome on the first spinner. We’re told that the spinners are numbered from one to four. So the outcome we’re missing here is four. For the second spinner, the outcome that we’re missing is the outcome of three. So we can fill in these two cells.
One of the numbers inside the table has already been filled in for us, this number five, which is indeed the sum of four on the first spinner and one on the second. We’ll now fill in the remaining cells in the table, remembering that we add together the scores on the two spinners each time.
A one on the first spinner and a one on the second spinner have a sum of two. A two on the first spinner and a one on the second spinner have a sum of three. And finally, for this row, a three on the first spinner and a one on the second have a sum of four. We then fill in the rest of the table in the same way.
Notice that there are some patterns in our table. Numbers that are the same are arranged in diagonals. For example, all of the fives are in the diagonal going from the bottom left to the top right. All of the fours are above this diagonal, and all of the sixes are below this diagonal. Above the fours, there are the threes. And below the sixes, there are the sevens. So here we have our completed table or completed sample space diagram. And we’ve answered part a) of the question.
Now let’s consider part b), which asked us to find the probability of getting a score that is a combined total for the numbers of this first spinner and the second, which is a square number.
A square number is the result of multiplying an integer by itself. For example, one is a square number because one multiplied by one is equal to one. Four is also a square number because two multiplied by two is equal to four. The next square number is nine because three multiplied by three is equal to nine.
Let’s have a look at our table and see if we can identify the square numbers. The only square number which appears in our table is four. And we can see that it appears three times, these three cells that I’ve highlighted in orange. How do we use this though to determine the probability of getting a square number?
Well, the probability can be found by dividing the number of successful outcomes, which in this question is the number of square numbers, by the total number of outcomes. We’ve just worked out that the number of square numbers in the table is three. But what about the total number of outcomes?
Well, on the inside of our table, there are four rows and four columns, which means there are four times four — that’s 16 — possible outcomes overall. You can add them up individually if you wish. But it’s much easier to work it out as four times four. So the total number of outcomes is 16. And we find that the probability of getting a score that is a square number is three over 16.
We can’t simplify this probability any further as there are no common factors in the numerator and denominator. And there’s no need to try and convert it to a decimal. So we’ve completed the table to answer part a). And then in part b), we found that the probability of getting a score that is a square number is three sixteenths.