The nine regions on a spinner are numbered with positive integers 1 through 9, and the probability of the spinner landing on a given region is proportional to the label for that region. For example, since $4 \times 2 = 8$, the probability of landing on 8 is twice the probability of landing on 4. What is the probability that this spinner will land on an odd number? Express your answer as a common fraction.
odd sum ... 1 + 3 + 5 + 7 + 9 = 25
even sum ... 2 + 4 + 6 + 8 = 20
p(odd) = 25 / (25 + 20)
let the area of the whole spinner be x
fraction of areas:
1 ---- 1/x
2 ---- 2/x
...
9 ---- 9/x
total area = 45/x
area of odds = 1/x + 3/x + 5/x + 7/x + 9/x = 25/x
prob(odd) = (25/x) / (45/x) = 25/45 = 5/9
Why did the spinner join a comedy club? Because it wanted to improve its "odd"-ience!
Now let's calculate the probability of landing on an odd number.
We have 5 odd numbers on the spinner: 1, 3, 5, 7, and 9.
The sum of all the numbers on the spinner is 1+2+3+4+5+6+7+8+9 = 45. And since the probability of each region is proportional to its label, the sum of the probabilities of landing on each region is also 45.
Since the probability of landing on each region is proportional to its label, the probability of landing on an odd number is the sum of the probabilities of landing on each odd number.
The sum of the probabilities of landing on odd numbers is 1 + 3 + 5 + 7 + 9 = 25.
Therefore, the probability of landing on an odd number is 25/45 which simplifies to 5/9.
So the probability that the spinner will land on an odd number is $\boxed{\frac{5}{9}}$.
To find the probability that the spinner will land on an odd number, we first need to determine the total number of possibilities and then find the number of favorable outcomes.
The spinner has nine regions numbered from 1 to 9. Since the probability of landing on a specific region is proportional to its label, we can determine the probability of landing on each number by multiplying it by its label.
The total number of possibilities is the sum of the labels of all nine regions:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45
Now let's find the number of favorable outcomes, which in this case are the odd numbers. The odd numbers on the spinner are 1, 3, 5, 7, and 9.
The probability of landing on an odd number can be found by summing the labels of these odd numbers:
1 + 3 + 5 + 7 + 9 = 25
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possibilities:
Probability = Number of favorable outcomes / Total number of possibilities
= 25 / 45
= 5 / 9
Therefore, the probability that the spinner will land on an odd number is 5/9.
To find the probability that the spinner will land on an odd number, we need to determine the sum of the probabilities of landing on all odd numbers and divide it by the sum of the probabilities of landing on all numbers.
To calculate the sum of the probabilities of landing on all odd numbers, we add up the probabilities of landing on the odd numbers 1, 3, 5, 7, and 9. The probabilities for these odd numbers are in proportion to their labels. So the probability of landing on 1 is 1 times a constant, the probability of landing on 3 is 3 times the same constant, and so on. Let's call this constant k.
The sum of the probabilities of landing on the odd numbers is:
1k + 3k + 5k + 7k + 9k = 25k.
Next, let's calculate the sum of the probabilities of landing on all numbers. Since the spinner has 9 regions and the probabilities are proportional to the labels, we can think of this sum as 1k + 2k + ... + 9k. This can be simplified by using the formula for the sum of an arithmetic series, which is given by (first term + last term) * number of terms / 2. In this case, the first term is 1k, the last term is 9k, and the number of terms is 9. Using the formula, we find that the sum of the probabilities of landing on all numbers is:
(1k + 9k) * 9 / 2 = 10k * 9 / 2 = 45k.
Finally, to find the probability of landing on an odd number, we divide the sum of the probabilities of landing on odd numbers by the sum of the probabilities of landing on all numbers:
(25k) / (45k) = 25/45 = 5/9.
Therefore, the probability that the spinner will land on an odd number is 5/9.