Question

the probability of event a, given that event b has occurred, can be found using bayess theorem.

\\p(a \mid b) = \frac{p(a) \cdot p(b \mid a)}{p(a) \cdot p(b \mid a) + p(a) \cdot p(b \mid a)}\\

use bayess theorem to find \\(p(a \mid b)\\) using the probabilities shown below.

\\(p(a) = 0.35\\), \\(p(a) = 0.65\\), \\(p(b \mid a) = 0.1\\), and \\(p(b \mid a) = 0.5\\)

the probability of event a, given that event b has occurred, is .
(round to the nearest thousandth as needed.)

Explanation:

Identify the given probabilities

Using the Conditional Probability Notation knowledge point, we extract the values from the problem statement:

$$
LATEXBLOCK0
$$

Calculate the joint probabilities

Using the Joint Probability Calculation knowledge point:

$$
LATEXBLOCK1
$$

Calculate total probability of event B

We sum the joint probabilities to find the total probability of the conditioning event:

$$ P(B) = P(A \cap B) + P(A' \cap B) = 0.035 + 0.325 = 0.360 $$

Apply Bayes's Theorem

We substitute the calculated values into the formula to find the posterior probability:

$$ P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{0.035}{0.360} $$

Simplify and round the final value

We perform the division and round to the nearest thousandth:

$$ P(A \mid B) = \frac{35}{360} \approx 0.097222\dots \approx 0.097 $$

Answer:

The probability of event A, given that event B has occurred, is <blank>0.097</blank>.