Suppose two spinners are spun. The first spinner is numbered from one to three and the second from one to nine. Using a tree diagram, determine what the probability of the sum of both spins being greater than four is.
The first thing we notice is that our question is asking us to use a tree diagram to solve this problem. We can start by creating archery diagram. Spinner number one is numbered from one to three. And so our first part of the tree diagram will look like this, a tree with three branches for our three options: one, two, and three. Our second spinner is numbered from one to nine. But this is a second event which means from each branch of our first event, we’ll need to label one to nine, like this. This branch shows all the cases if a one is spun on the first spinner. We can have a one and then a one, one and then a two, one and then a three, and so on, all the way up until nine. But we need to repeat this process for branch two and branch three. Now on to branch three.
Now that we’ve successfully created our tree diagram, let’s go back and have a closer look at what the question is asking. Our question is asking us to determine the probability that the sum of both spins is greater than four. Right now, what we need to do is we need to determine what the sum of every spin would be.
Following our diagram down branch one, if we spun a one and then a one, we would have a sum of two. Our next branch, if we spun a one and then a two, we would get three. One and then three gives us four. One and then four yields the sum of five. One plus five is six. One plus six is seven. One plus seven is eight. One plus eight equals nine. One plus nine equals 10. We could move on to our second branch. Our second branch will be two plus all the values from one to nine. Two plus one is three, two plus two is four, and we’ll follow the same process all the way around. Two plus four is six, two plus five is seven, two plus six is eight, two plus seven is nine, two plus eight is 10, and two plus nine equals 11. We’ll do this process one final time for the three branch. And from left to right, we would get four, five, six, seven, eight, nine, 10, 11, 12 as our sums.
From here, we’ll go through all of our sums and we’ll circle the ones that are greater than four. If we add up all the values that we circled, we find that there are 21 values greater than four, 21 sums that are greater than four. Remember that to find probability here, we’ll be looking for the number of times that we have something greater than four over the total outcomes. The outcome we’re looking for, greater than four, over the total outcomes. We know that there are 21 times when the sum is greater than four. And if we add all the possible outcomes, there are 27 options, there are three times nine options. The probability that the sum of both spins are greater than four is 21 over 27. We could also list the probability as 21 out of 27.
21 of the times out of a total of 27 times, the sum of both spins is greater than four.