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.
What is the probability that I stop at neither of the traffic lights on my commute?
To find the probability that you stop at neither of the traffic lights (A'∩B'), we can use the formula:
P(A'∩B') = 1 - P(A∪B)
First, let's find P(A∪B) using the formula for the union of two events:
P(A∪B) = P(A) + P(B) - P(A∩B)
P(A∪B) = 0.4 + 0.5 - 0.3
P(A∪B) = 0.6
Now, we can find P(A'∩B'):
P(A'∩B') = 1 - P(A∪B)
P(A'∩B') = 1 - 0.6
P(A'∩B') = 0.4
Therefore, the probability that you stop at neither of the traffic lights on your commute is 0.4 or 40%.
To calculate the probability that you stop at neither of the traffic lights on your commute, we can use the formula:
P(A'∩B') = 1 - P(A∪B)
P(A∪B) represents the probability that you stop at either of the traffic lights or both.
Given that:
P(A) = 0.4,
P(B) = 0.5,
P(A∩B) = 0.3,
We can calculate:
P(A∪B) = P(A) + P(B) - P(A∩B)
P(A∪B) = 0.4 + 0.5 - 0.3
P(A∪B) = 0.6
Now, we can substitute this value into the formula to find P(A'∩B'):
P(A'∩B') = 1 - P(A∪B)
P(A'∩B') = 1 - 0.6
P(A'∩B') = 0.4
Therefore, the probability that you stop at neither of the traffic lights on your commute is 0.4 or 40%.
To find the probability that you stop at neither of the traffic lights, we need to calculate the complement probability of both events A and B.
Let's break down the problem using the given information:
P(A) = Probability you stop at the first traffic light = 0.4
P(B) = Probability you stop at the second traffic light = 0.5
P(A∩B) = Probability you stop at both traffic lights = 0.3
To find the probability of not stopping at the first traffic light (not A), we subtract P(A) from 1:
P(not A) = 1 - P(A)
P(not A) = 1 - 0.4
P(not A) = 0.6
Similarly, to find the probability of not stopping at the second traffic light (not B), we subtract P(B) from 1:
P(not B) = 1 - P(B)
P(not B) = 1 - 0.5
P(not B) = 0.5
Now, we need to find the probability of not stopping at both traffic lights, which is the complement of the intersection of events A and B (A∩B). We can use the formula:
P(not A∩B) = 1 - P(A∩B)
P(not A∩B) = 1 - 0.3
P(not A∩B) = 0.7
Since we want to find the probability of stopping at neither of the traffic lights, we multiply the individual probabilities of not stopping at each light:
P(not A and not B) = P(not A) * P(not B)
P(not A and not B) = 0.6 * 0.5
P(not A and not B) = 0.3
Therefore, the probability that you stop at neither of the traffic lights on your commute is 0.3 or 30%.