Video: Using Tree Diagrams to Determine the Probability of the Intersection of Two Events Involving Spinners

Suppose two spinners are spun. The first has 5 equal sectors numbered from 1 to 5, and the second has 9 equal sectors numbered from 1 to 9. Using a tree diagram or otherwise, find the probability that both spinners stop at odd numbers.

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Video Transcript

Suppose two spinners are spun. The first has five equal sectors numbered from one to five. And the second has nine equal sectors numbered from one to nine. Using a tree diagram or otherwise, find the probability that both spinners stop at odd numbers.

We can answer this question using a tree diagram. A tree diagram allows us to model different outcomes along with the probabilities of those outcomes. As our question is asking for odd numbers, let’s use our tree diagram to model the odd and the even numbers in both spinners. We can use the first set of branches to represent our first spinner. We’re told that our first spinner has the numbers one to five. And as these are equal sections, then the probability of each number is equal.

To calculate the probability of getting an odd number, then we simply calculate how many odd numbers that are on this spinner. As there are three odd numbers, then our probability will be three out of five or three-fifths. The probability of not getting an odd number, that is, getting an even number then, we count the even numbers. Since two and four are the even numbers, then we have two out of five or two-fifths.

We can now model the outcomes for our second spinner. The probability of getting an odd number in the top branches will be the same as it is in the bottom branches. And the same is true for the even probability. This is because the outcome of the first spinner does not affect the outcome of the second spinner.

So now, to calculate the probability of odd on our second spinner, we know that we have the numbers one to nine. Therefore, our odd values will be one, three, five, seven, and nine, giving us a probability of five-ninths. The probability of getting an even number will be the remaining four numbers out of nine. The probability of getting an odd and an even number in the lower section will be the same as those in the top section.

And now, to calculate the probability that both spinners stop at odd numbers, then we follow the path of getting an odd number on the first spinner and an odd number on the second spinner. And when we move along branches on a tree diagram, we multiply the probabilities. We can recall that when we multiply fractions, we multiply the numerators and multiply the denominators. So three-fifths times five-ninths will give us 15 over 45. And this is our final answer for the probability that both spinners stop at odd numbers.

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