QuestionAn that both A and B will occur is 0.1 .
8. The conditional probability of A , given B
(a) is .
8. The conditional probability of A , given B
(a) is .
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ur is ility 0.5 . An event will occur with probability 0.6 . The probability
(b) is .
(c) is .
(d) is .
(1) cannot be determined from the information given.
Studdy Solution
STEP 1
What is this asking? Given the probability of event A, the probability of event B, and the probability that both A and B occur, what's the probability of A happening *if* we know B already happened? Watch out! Don't mix up the probability of both events happening with the probability of one event happening *given* the other event already happened!
STEP 2
1. Define the probabilities
2. Apply the formula for conditional probability
3. Calculate the result
STEP 3
We're given , which is the probability of event A happening.
Awesome!
STEP 4
We also know , the probability of event B happening.
Fantastic!
STEP 5
And, importantly, we have , the probability that *both* A and B happen.
Super! This is what makes this problem interesting.
STEP 6
The **conditional probability** formula is what we need!
It tells us the probability of A happening *given* that B has already happened.
It's written as and is calculated as:
This formula is magical!
It tells us how the probability of A changes when we know B has already occurred.
The numerator is the probability of *both* A and B happening, and the denominator is the probability of B happening.
STEP 7
Let's plug in our **known values**: and .
This gives us:
STEP 8
Now, let's **simplify the fraction** by multiplying the numerator and denominator by **10** to get rid of the decimals:
So, the probability of A happening given that B has already happened is .
Perfect!
STEP 9
The conditional probability of A given B is , which corresponds to answer (d).
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