Question

a spinner has 12 equally sized slices numbered from 1 to 12. some are white and some are grey, as shown below. the slices numbered 2, 4, 6, 8, 9, and 10 are white. the slices numbered 1, 3, 5, 7, 11, and 12 are grey. answer the following questions. write each answer as a fraction. the wheel will be spun and will stop on a slice at random. (a) what is the probability that the wheel stops on number 9? (b) what is the probability that the wheel stops on number 9, given that the wheel stops on a white slice?

Explanation:

Part (a)

Step1: Determine total outcomes

There are 12 equally - sized slices, so the total number of possible outcomes \(n = 12\).

Step2: Determine favorable outcomes

We want the wheel to stop on number 9. There is only 1 slice with the number 9, so the number of favorable outcomes \(m=1\).

Step3: Calculate probability

The probability \(P\) of an event is given by the formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). So \(P(\text{stop on 9})=\frac{1}{12}\).

Step1: Determine the number of white slices

The white slices are numbered 2, 4, 6, 8, 9, and 10. So the number of white slices \(n(\text{white}) = 6\).

Step2: Determine the number of white slices with number 9

There is only 1 white slice with the number 9, so \(n(\text{white and 9})=1\).

Step3: Calculate conditional probability

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In the case of equally - likely outcomes, it can also be calculated as \(P(\text{9 }|\text{ white})=\frac{\text{Number of white slices with 9}}{\text{Number of white slices}}\). Substituting the values, we get \(P(\text{9 }|\text{ white})=\frac{1}{6}\).

Answer:

\(\frac{1}{12}\)

Part (b)