Ms. Tucker travels through two intersections with traffic lights as she drives to the market. The traffic lights operate independently. The probability that both lights will be red when she reaches them is .22. the probability that the first light will be red and the second light will not be red is .33. what is the probability that the second light will be red when she reaches it?
.4
The probability of two independent events occurring in succession is the product of the two events' independent probabilities. .22 = that product. Because we know the probability of the first being red and the second not being red (.33), we know the probability of the first light being red: .22 + .33. That comprises all the possible ways the first light can be red.
From there, just divide .22 by .55 in order to get .4. Tough problem because they phrased it ambiguously.
You need a tree diagram. Make the branch Red Red =.22, and Red NRed=.33 and assign variables on the starting branches, then solve for the variable values.
To solve this problem, we need to use conditional probability. Let's denote the events as follows:
A: The first light is red
B: The second light is red
We are given the following probabilities:
P(A and B) = 0.22 (the probability that both lights are red)
P(A and not B) = 0.33 (the probability that the first light is red and the second light is not red)
We want to find P(B) (the probability that the second light is red).
We can begin by using the concept of conditional probability, which states:
P(A and B) = P(B | A) * P(A)
Rearranging the equation, we can solve for P(B):
P(B) = P(A and B) / P(A)
Now let's substitute the given values:
P(B) = 0.22 / P(A)
To find P(A), we can use the law of total probability, which states that the probability of an event can be calculated by considering the probabilities of its various mutually exclusive and exhaustive possibilities. In this case, there are two possibilities:
1) The first light is red and the second light is red.
2) The first light is red and the second light is not red.
When we sum the probabilities of these two possibilities, we should get 1:
P(A) + P(A and not B) = 1
Given that P(A and not B) = 0.33, we can solve for P(A):
P(A) = 1 - P(A and not B)
= 1 - 0.33
= 0.67
Now we have enough information to find P(B):
P(B) = 0.22 / 0.67
Calculating the value:
P(B) ≈ 0.328 (rounded to three decimal places)
Therefore, the probability that the second light will be red when Ms. Tucker reaches it is approximately 0.328 or 32.8%.