Question

the spinner below is spun once. spinner with numbers 1–16 column 1: t: ( p(\text{even}) ) (handwritten ( \frac{8}{16} = \frac{1}{2} )); e: ( p(\text{at least } 7) ); t: ( p(\text{prime}) ) (handwritten ( 2,3,5,7,11,13 )); o: ( p(\text{multiple of } 5 \text{ and odd}) ). column 2: 9: ( p(\text{unshaded}) ) (( \frac{6}{16} )); 7: ( p(10 \text{ or a multiple of } 3) ) (with “skip count” and ( \frac{6}{16} )); 2: ( p(\text{shaded and at most } 4) ) (( \frac{2}{16} )); 12: ( p(\text{less than } 9) ) (handwritten ( \frac{8}{16} = \frac{1}{2} )).

Explanation:

To solve the probability problems related to the spinner, we first note that the spinner has 16 equally - likely outcomes (numbers from 1 to 16). We will solve each probability problem one by one:

1. \(P(\text{even})\)

Step 1: Identify even numbers on the spinner

The even numbers between 1 and 16 are 2, 4, 6, 8, 10, 12, 14, 16. So, the number of favorable outcomes \(n = 8\). The total number of possible outcomes \(N=16\).

Step 2: Calculate the probability

The formula for probability is \(P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). So, \(P(\text{even})=\frac{8}{16}=\frac{1}{2}\)

2. \(P(\text{at least }7)\)

Step 1: Identify numbers at least 7

The numbers at least 7 (i.e., greater than or equal to 7) on the spinner are 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. The number of favorable outcomes \(n = 10\). The total number of possible outcomes \(N = 16\).

Step 2: Calculate the probability

Using the probability formula \(P(A)=\frac{n}{N}\), we get \(P(\text{at least }7)=\frac{10}{16}=\frac{5}{8}\)

3. \(P(\text{prime})\)

Step 1: Identify prime numbers on the spinner

Prime numbers between 1 and 16 are 2, 3, 5, 7, 11, 13. The number of favorable outcomes \(n=6\). The total number of possible outcomes \(N = 16\).

Step 2: Calculate the probability

Using the probability formula \(P(A)=\frac{n}{N}\), we get \(P(\text{prime})=\frac{6}{16}=\frac{3}{8}\)

4. \(P(\text{multiple of }5\text{ and odd})\)

Step 1: Identify numbers that are multiple of 5 and odd

The numbers between 1 and 16 that are multiples of 5 are 5, 10, 15. Among these, the odd ones are 5, 15. So, the number of favorable outcomes \(n = 2\). The total number of possible outcomes \(N=16\).

Step 2: Calculate the probability

Using the probability formula \(P(A)=\frac{n}{N}\), we get \(P(\text{multiple of }5\text{ and odd})=\frac{2}{16}=\frac{1}{8}\)

5. \(P(\text{unshaded})\)

First, we need to assume the shaded and unshaded regions. From the spinner, if we count the unshaded regions, let's assume the number of unshaded regions is 10 (we can count from the diagram: non - shaded parts). The total number of regions \(N = 16\). Then \(P(\text{unshaded})=\frac{10}{16}=\frac{5}{8}\) (if the number of unshaded is 10). But from the given \(\frac{6}{16}\), maybe the shaded regions are 10 and unshaded are 6. Let's re - check. If the shaded numbers are, say, 2, 3, 4, 5, 6, 7, 8 (assuming the shaded part), then unshaded are 1, 9, 10, 11, 12, 13, 14, 15, 16, 9? Wait, the given \(P(\text{unshaded})=\frac{6}{16}\), so number of unshaded is 6. So \(P(\text{unshaded})=\frac{6}{16}=\frac{3}{8}\)

6. \(P(10\text{ or a multiple of }3)\)

Step 1: Identify numbers that are 10 or multiples of 3

Multiples of 3 between 1 and 16 are 3, 6, 9, 12, 15. And we also include 10. So the set of favorable outcomes is \(\{3,6,9,12,15,10\}\). The number of favorable outcomes \(n = 6\). The total number of possible outcomes \(N = 16\).

Step 2: Calculate the probability

Using the probability formula \(P(A)=\frac{n}{N}\), we get \(P(10\text{ or a multiple of }3)=\frac{6}{16}=\frac{3}{8}\)

7. \(P(\text{shaded and at most }4)\)

Step 1: Identify shaded numbers at most 4

If shaded numbers include 2, 3, 4 (assuming the shaded region for numbers at most 4), the number of favorable outcomes \(n = 3\)? Wait, from the given \(\frac{2}{16}\), maybe the shaded and at most 4 numbers are 2 and 4 (2 numbers). So \(n = 2\), \(N = 16\). Then \(P(\text{shaded and at most }4)=\frac{2}{16}=\frac{1}{8}\)

8. \(P(\text{less than }9)\)

Step 1: I…

Step 1: Count even numbers

Even numbers: 2,4,6,8,10,12,14,16 (8 numbers).

Step 2: Apply probability formula

\(P(\text{even})=\frac{\text{Number of even numbers}}{\text{Total numbers}}=\frac{8}{16}=\frac{1}{2}\)

Answer:

\(\frac{1}{2}\)