On my commute to work, I pass through two intersections with traffic lights. Based on a large amount of empirical data, I estimate that:
- the probability I stop at the first traffic light (event A) is 0.4
- the probability I stop at the second traffic light (event B) is 0.5,
- and the probability that I stop at both traffic lights (A∩B) is 0.3.
Given that I stop at the first traffic light, what is the probability that I also stop at the second?
To find the probability that you stop at the second traffic light given that you stop at the first traffic light, you need to use conditional probability. The formula for conditional probability is:
P(A|B) = P(A∩B) / P(A)
In this case, we want to find P(B|A), which is the probability of stopping at the second traffic light given that you stop at the first traffic light. Plugging in the given values:
P(B|A) = P(A∩B) / P(A)
P(B|A) = 0.3 / 0.4
P(B|A) = 0.75
Therefore, the probability that you stop at the second traffic light given that you stop at the first traffic light is 0.75 or 75%.
To find the probability that you stop at the second traffic light given that you stop at the first traffic light, you can use conditional probability.
Let's define:
- P(A) as the probability of stopping at the first traffic light
- P(B) as the probability of stopping at the second traffic light
- P(A∩B) as the probability of stopping at both traffic lights
We want to find P(B|A), which represents the probability of stopping at the second traffic light given that you stop at the first traffic light.
Using conditional probability formula:
P(B|A) = P(A∩B) / P(A)
Plugging in the values:
P(B|A) = 0.3 / 0.4
Simplifying:
P(B|A) = 0.75
Therefore, the probability that you stop at the second traffic light given that you stop at the first traffic light is 0.75 or 75%.
To find the probability that you stop at the second traffic light (B) given that you stop at the first traffic light (A), we can use conditional probability.
Conditional probability is used when you have information about one event and want to find the probability of another event under that given information.
Mathematically, the conditional probability of event B given event A is denoted as P(B|A), which reads as "probability of event B given event A."
In this case, we want to find P(B|A), which represents the probability of stopping at the second traffic light given that we have already stopped at the first traffic light.
The formula for conditional probability is:
P(B|A) = P(A∩B) / P(A)
Given the information you provided:
- P(A∩B) is 0.3 (the probability of stopping at both traffic lights)
- P(A) is 0.4 (the probability of stopping at the first traffic light)
We can substitute these values into the conditional probability formula:
P(B|A) = P(A∩B) / P(A)
= 0.3 / 0.4
Now we can calculate the probability:
P(B|A) = 0.3 / 0.4
= 0.75
= 75%
Therefore, the probability that you stop at the second traffic light given that you stop at the first traffic light is 0.75 or 75%.