Let's break down the problem shown in the image:

We have a dartboard divided into 8 equally sized slices, numbered from 1 to 8. Some of the slices are grey and some are white. The task is to compute two probabilities:

Part (a):

The first question asks:

• What is the probability that the dart lands on a grey slice?

We can count the number of grey slices from the diagram to determine the favorable outcomes.

Part (b):

The second question asks:

• What is the probability that the dart lands on a grey slice, given that the dart lands on an even-numbered slice?

This is a conditional probability problem. The probability of landing on a grey slice, given that the slice is even-numbered, can be calculated by considering the even-numbered slices first and checking how many of them are grey.

I'll count the slices from the image and solve these probabilities for you.From the image, we can see the dartboard has 8 equally sized slices numbered from 1 to 8, and the slices alternate between grey and white.

Step 1: Counting the grey slices

• Grey slices: 1, 3, 5, 7 There are 4 grey slices.

Step 2: Solving Part (a)

The total number of slices is 8, and there are 4 grey slices. Therefore, the probability that the dart lands on a grey slice is:

$P(\text{grey slice}) = \frac{4}{8} = \frac{1}{2}$

Step 3: Solving Part (b)

Now, we need to find the probability that the dart lands on a grey slice, given that it lands on an even-numbered slice.

• Even-numbered slices: 2, 4, 6, 8 Out of these, none of them are grey; they are all white.

Thus, the probability that the dart lands on a grey slice given that it lands on an even-numbered slice is:

$P(\text{grey | even}) = 0$

Final Answers:

• (a) The probability that the dart lands on a grey slice is $\frac{1}{2}$.
• (b) The probability that the dart lands on a grey slice given that it lands on an even-numbered slice is $0$.

Would you like further clarification on these? Also, here are 5 related questions:

1. What is the probability of landing on an odd-numbered slice?
2. How would the probabilities change if there were 10 slices instead of 8?
3. What if every third slice is grey? What would the probability be of landing on a grey slice?
4. What is the probability of landing on a white slice?
5. How can conditional probability be applied to real-life situations involving dice?

Tip: Conditional probabilities are especially useful in situations where you already have some known information (like "the dart landed on an even-numbered slice") and want to calculate the probability based on that condition.