Finding conditional probability of running on a windy day

Expert intro: This is a conditional probability question: we are asked for the chance of running under the condition that the day is windy.

Detailed walkthrough

Identify the conditional probability

Let $W$ be the event that the day is windy and $R$ be the event that you go for a run. The question asks for $P(R\mid W)$, which means “the probability of running given that it is windy.” In the problem, that value is stated directly.

The probability of a windy day is $P(W)=52\%$, and the probability of going for a run when it is windy is $P(R\mid W)=23\%$. Since the condition is already included in the statement, no extra calculation is needed.

Apply the definition

Conditional probability describes the chance of one event happening after another event has already been fixed. Here, the fixed condition is windy weather.

So the required probability is

$

$P(R\mid W)=23\%=0.23$

$

Check the wording carefully

A common mistake is to multiply $52\%$ and $23\%$ without reading the question type. That multiplication would find the probability of both events together, $P(R\cap W)$, not the conditional probability asked here.

If the question had asked for the probability of both a windy day and a run, then the calculation would be $0.52\times 0.23=0.1196$, or $11.96\%$. But that is not the requested value.

Final answer

The probability of going on a run, given that it is a windy day, is 23%.

💡 Pitfall guide

The biggest trap is confusing conditional probability with joint probability. If you see words like “given that,” the answer is usually the probability inside the condition, not a multiplication with the condition’s probability. Students also sometimes reverse the condition and write $P(W\mid R)$ instead of $P(R\mid W)$. Those are different quantities and can have very different values. Always identify the event after the bar first.

🔄 Real-world variant

If the question were changed to “What is the probability of a windy day and going for a run?” then you would use multiplication: $P(W\cap R)=P(W)\cdot P(R\mid W)=0.52\times 0.23=0.1196$, or $11.96\%$. If it asked for “the probability of running on a non-windy day,” you would need a different conditional probability, which is not provided here. That is why the exact wording matters.

🔍 Related terms

conditional probability, joint probability, event notation