Calculate the probabilities of every possible sum of two spins.
Liana got it right, see solution.
Practice makes perfect
To determine who is right, Liana or Shanon, we will make a probability distribution graph ourselves and compare the results. First, let's look at the spinner.
We can see that 4 takes up half of the spinner. Therefore, the probability of the spinner stopping at 4 is 12. We can do similar observations with the other numbers. Let's write down all the numbers and their corresponding probabilities in a table.
Value of a Single Spin
Probability
2
1/8
3
1/4
4
1/2
5
1/8
Now, we will write the ways in which we can get every possible sum. The possible sums are the numbers from 4 to 10.
Possible Sum of Two Spins
Two Spins
4
2+2
5
2+3,3+2
6
3+3,2+4,4+2
7
3+4,4+3,2+5,5+2
8
4+4,3+5,5+3
9
4+5,5+4
10
5+5
Next, we can find the probabilities of the possible sums of two spins. For example, let's calculate the probability that the sum of spins is equal to 5. To do that, we need to find the probabilities that one of the spins was equal to 2 and the other was 3.
5=2+3 or 5=3+2
Since the spins are independent, the probability that one of the spins was 2 and the other was 3 is the probability of spinning 2 multiplied by the probability of spinning 3.
P(First Spin2,Second3)=P(2)* P(3)
P(First Spin3,Second2)=P(3)* P(2)
We can substitute the values of probabilities P(2) and P(3) from the first table we made in this exercise.
P(First Spin2,Second3)=1/4*1/8= 1/32
P(First Spin3,Second2)=1/8*1/4= 1/32
Now, to find the probability that the sum of two spins is equal to 5 we need add up the probabilities of every possible way that the sum of two spins was equal to 5. Which, in this case means that we need to add the probabilities P(First Spin2,Second3) and P(First Spin3,Second2).
P(Two Spins Equal5)= 1/32+ 1/32=1/16
Looking at this result, we can already tell that Shannon's solution is different. She thinks that this probability is equal to only 132, which is only half of what we got. This suggests that she did not consider both situations of getting 5 as the sum of two spins. Now that we know that Shannon is wrong, let's continue calculating the probabilities to see if Liana got it right.
Possible Sum of Two Spins
Two Spins
Probability
4
2+2
1/64
5
2+3,3+2
1/16
6
3+3,2+4,4+2
3/16
7
3+4,4+3,2+5,5+2
9/32
8
4+4,3+5,5+3
5/16
9
4+5,5+4
1/8
10
5+5
1/64
Finally, we can make a probability distribution graph and check if it matches the one that Liana made. Every possible sum of two spins will be represented by a bar. The bar's height should be proportional to the corresponding probability.
Our result is the same as Liana's! Therefore, she must be right.
