Questions: Given P(A B)=0.3 and P(B)=0.65 find P(A and B). Round your answer to four decimals. P(A and B)=

Transcript text: Given $P(A \mid B)=0.3$ and $P(B)=0.65$ find $P(A$ and $B)$. Round your answer to four decimals. \[ P(A \text { and } B)= \]

Solution

Solution Steps

Step 1: Understand the given probabilities

We are given:

$ P(A \mid B) = 0.3 $, which is the probability of event $ A $ occurring given that event $ B $ has occurred.
$ P(B) = 0.65 $, which is the probability of event $ B $ occurring.

Step 2: Recall the formula for conditional probability

The formula for conditional probability is: \[ P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)} \] We can rearrange this formula to solve for $ P(A \text{ and } B) $: \[ P(A \text{ and } B) = P(A \mid B) \cdot P(B) \]

Step 3: Substitute the given values into the formula

Substitute $ P(A \mid B) = 0.3 $ and $ P(B) = 0.65 $ into the formula: \[ P(A \text{ and } B) = 0.3 \cdot 0.65 \]

Step 4: Calculate the result

Multiply the values: \[ P(A \text{ and } B) = 0.195 \]

Step 5: Round the result to four decimal places

The result $ 0.195 $ is already to three decimal places. To round it to four decimal places, we add a zero: \[ P(A \text{ and } B) = 0.1950 \]

\[ P(A \text{ and } B) = \boxed{0.1950} \]