Question

1. an individual is chosen from the group. according to the venn diagram,

1a what is the probability that the person walks?
= \frac{17}{24}
well done!
1b if the person chosen has been recorded as walking, what is the probability that they also run?
= \frac{9}{9 + 8}
explain
= \frac{9}{9 + 8}

Explanation:

Step1: Identify the context (conditional probability)

We need to find the probability that a person runs given that they walk. This is a conditional probability problem, denoted as \( P(\text{run} | \text{walk}) \). The formula for conditional probability is \( P(A|B)=\frac{P(A\cap B)}{P(B)} \), which in terms of counts (since we can use the number of people) is \( \frac{\text{Number of people who walk and run}}{\text{Number of people who walk}} \).

Step2: Determine the counts

From the Venn diagram (implied by the given \( 9 + 8 \) for the denominator of the walkers, and 9 for those who walk and run), the number of people who walk and run is 9, and the number of people who walk is the number of people who walk only plus those who walk and run, which is \( 9+8 \) (assuming 8 are walk - only and 9 are walk - and - run). So the number of walkers is \( 9 + 8=17 \) (wait, but in part 1a, the probability of walking was \( \frac{17}{21} \), so total walkers are 17). The number of people who walk and run is 9.

Step3: Apply the conditional probability formula

Using the formula \( P(\text{run}|\text{walk})=\frac{\text{Number of walk and run}}{\text{Number of walk}} \), substituting the values, we get \( \frac{9}{9 + 8}=\frac{9}{17} \). Wait, the initial wrong answer had \( \frac{9}{9 + 8} \) but let's check again. Wait, if the number of people who walk is the sum of those who walk only and those who walk and run. If 9 are in the intersection (walk and run) and 8 are only walk, then total walkers are \( 9+8 = 17 \). So the probability that a walker also runs is \( \frac{9}{17} \). But let's re - express the steps:

1. Recall the formula for conditional probability: \( P(\text{Run}|\text{Walk})=\frac{n(\text{Walk}\cap\text{Run})}{n(\text{Walk})} \)
2. \( n(\text{Walk}\cap\text{Run}) = 9 \) (from the Venn diagram, the number of people who do both)
3. \( n(\text{Walk})=n(\text{Walk only})+n(\text{Walk}\cap\text{Run}) \). From part 1a, \( P(\text{Walk})=\frac{17}{21} \), so \( n(\text{Walk}) = 17 \) (since total number of people is 21, as \( \frac{17}{21} \) was the probability of walking). Wait, maybe the 8 is a miscalculation. Wait, if total walkers are 17, and those who walk and run are 9, then walk - only is \( 17-9 = 8 \). So the number of walkers is 17, and the number of walk - and - run is 9. So \( P(\text{run}|\text{walk})=\frac{9}{17} \). But the initial expression \( \frac{9}{9 + 8} \) is equal to \( \frac{9}{17} \) because \( 9+8 = 17 \). So the correct probability is \( \frac{9}{17} \), and the initial wrong answer had the correct expression for the fraction (but maybe a miscalculation in the thought process, but the formula \( \frac{9}{9 + 8} \) simplifies to \( \frac{9}{17} \)).

Answer:

The correct probability that a person who walks also runs is \( \frac{9}{17} \) (the initial expression \( \frac{9}{9 + 8} \) is correct in form, and when simplified, \( 9+8 = 17 \), so \( \frac{9}{17} \)). Wait, let's re - do:

If we use the formula for conditional probability \( P(A|B)=\frac{n(A\cap B)}{n(B)} \)

Let \( A=\text{run} \), \( B = \text{walk} \)

\( n(A\cap B) \): number of people who run and walk = 9

\( n(B) \): number of people who walk. From part 1a, \( P(B)=\frac{17}{21} \), so if we assume the total number of people is \( N \), then \( \frac{n(B)}{N}=\frac{17}{21} \). But from the Venn diagram, \( n(B)=n(\text{walk only})+n(A\cap B) \). If \( n(A\cap B) = 9 \) and \( n(\text{walk only})=8 \), then \( n(B)=9 + 8=17 \), so \( N = 21 \) (since \( \frac{17}{21} \) is the probability of walking).

So \( P(A|B)=\frac{n(A\cap B)}{n(B)}=\frac{9}{17} \), and \( \frac{9}{9 + 8}=\frac{9}{17} \) (because \( 9+8 = 17 \)). So the correct probability is \( \frac{9}{17} \), and the expression \( \frac{9}{9 + 8} \) is correct as a step (since \( 9+8 \) is the number of walkers), and when simplified, it is \( \frac{9}{17} \).