Your professor tells you that if you score an 85 or better on your midterm exam, then you have a 90% chance of getting an A for the course. You think you have only a 50% chance of scoring 85 or better. Find the probability that your score is 85 or better and you receive an A in the course.

If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.

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To find the probability that your score is 85 or better and you receive an A in the course, we can use conditional probability.

Let's define the events:
A: You receive an A in the course.
B: Your score is 85 or better on the midterm exam.

Given information:
P(A|B) = 0.90 (The probability of receiving an A given that your score is 85 or better)
P(B) = 0.50 (The probability of scoring 85 or better)

We want to find P(A ∩ B) which represents the probability of both events A and B occurring.

Using the formula for conditional probability:
P(A|B) = P(A ∩ B) / P(B)

We can rearrange the formula to solve for P(A ∩ B):
P(A ∩ B) = P(A|B) * P(B)

Substituting the given values into the equation:
P(A ∩ B) = 0.90 * 0.50

Calculating the result:
P(A ∩ B) = 0.45

Therefore, the probability that your score is 85 or better and you receive an A in the course is 0.45 or 45%.

To find the probability that your score is 85 or better and you receive an A in the course, we can use conditional probability. Let's break it down step by step:

Step 1: Given the information provided, we know the following probabilities:
- P(A | B): The probability of getting an A given that you score 85 or better.
- P(B): The probability of scoring 85 or better.
- P(A): The probability of receiving an A in the course.

Step 2: According to the information, P(B) = 0.50, since you think you have a 50% chance of scoring 85 or better.

Step 3: The professor states that if you score 85 or better, there is a 90% chance of getting an A. Mathematically, this is expressed as P(A | B) = 0.90.

Step 4: We need to find P(A ∩ B), which represents the probability of both scoring 85 or better and receiving an A. Since P(A ∩ B) is what we're looking for, let's assign this value as x.

Step 5: Using conditional probability, we know that P(A ∩ B) = P(A | B) * P(B). Therefore, we can write the equation as x = 0.90 * 0.50.

Step 6: Solve the equation to find x. Multiply 0.90 by 0.50 to get 0.45. Thus, x = 0.45.

Step 7: Therefore, the probability that your score is 85 or better and you receive an A in the course is 0.45 or 45%.

In summary, the probability is 45% that your score is 85 or better and you receive an A in the course.