Math / Data & Statistics

Question< 13 14 15 $=16$ $\equiv 17$ 18 $=19$
Let $A$ and $B$ be events with $P(A)=0.8, P(B)=0.6$ , and $P(B \mid A)=0.5$ . Find $P(A$ and $B)$ . $P(A \text { and } B)=$ $\square$

Studdy Solution

STEP 1

What is this asking? Given the probabilities of events A and B, and the probability of B happening given that A has already happened, what's the probability that both A and B happen? Watch out! Don't mix up $P(B \mid A)$ with $P(A \mid B)$ or with $P(A \text{ and } B)$ ! $P(B \mid A)$ is the probability of B happening after we know A has already happened.

STEP 2

1. Understand Conditional Probability
2. Calculate Joint Probability

STEP 3

Alright, so we're given $P(A) = \mathbf{0.8}$ , $P(B) = \mathbf{0.6}$ , and $P(B \mid A) = \mathbf{0.5}$ .
Remember, $P(B \mid A)$ means "the probability of B happening given that A has already happened." It's like saying, "If we know A happened, what are the chances of B also happening?"

STEP 4

The formula for conditional probability is super important: $P(B \mid A) = \frac{P(A \text{ and } B)}{P(A)}$ This tells us that the probability of B given A is the probability of both A and B happening, divided by the probability of A happening.

STEP 5

We want to find $P(A \text{ and } B)$ , the probability that both A and B happen.
We can rearrange the conditional probability formula to solve for this: $P(A \text{ and } B) = P(B \mid A) \cdot P(A)$ This makes sense, right?
The probability of both happening is the probability of A happening, times the probability of B happening given that A has already happened.

STEP 6

Now, we just plug in the values we know: $P(A \text{ and } B) = \mathbf{0.5} \cdot \mathbf{0.8}$ $P(A \text{ and } B) = \mathbf{0.4}$ So, the probability of both A and B happening is 0.4.

STEP 7

$P(A \text{ and } B) = 0.4$

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Question< 13 14 15 $=16$ $\equiv 17$ 18 $=19$
Let $A$ and $B$ be events with $P(A)=0.8, P(B)=0.6$ , and $P(B \mid A)=0.5$ . Find $P(A$ and $B)$ . $P(A \text { and } B)=$ $\square$

Studdy Solution

STEP 1

STEP 2

1. Understand Conditional Probability
2. Calculate Joint Probability

STEP 3

Alright, so we're given $P(A) = \mathbf{0.8}$ , $P(B) = \mathbf{0.6}$ , and $P(B \mid A) = \mathbf{0.5}$ .
Remember, $P(B \mid A)$ means "the probability of B happening given that A has already happened." It's like saying, "If we know A happened, what are the chances of B also happening?"

STEP 4

The formula for conditional probability is super important: $P(B \mid A) = \frac{P(A \text{ and } B)}{P(A)}$ This tells us that the probability of B given A is the probability of both A and B happening, divided by the probability of A happening.

STEP 5

STEP 6

Now, we just plug in the values we know: $P(A \text{ and } B) = \mathbf{0.5} \cdot \mathbf{0.8}$ $P(A \text{ and } B) = \mathbf{0.4}$ So, the probability of both A and B happening is 0.4.

STEP 7

$P(A \text{ and } B) = 0.4$

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Question< 13 14 15 =16=16=16 ≡17\equiv 17≡17 18 =19=19=19Let AAA and BBB be events with P(A)=0.8,P(B)=0.6P(A)=0.8, P(B)=0.6P(A)=0.8,P(B)=0.6, and P(B∣A)=0.5P(B \mid A)=0.5P(B∣A)=0.5. Find P(AP(AP(A and B)B)B). P(A and B)=P(A \text { and } B)=P(A and B)= □\square□

Studdy Solution

STEP 1

STEP 2

1. Understand Conditional Probability2. Calculate Joint Probability

STEP 3

STEP 4

STEP 5

STEP 6

Now, we just plug in the values we know: P(A and B)=0.5⋅0.8P(A \text{ and } B) = \mathbf{0.5} \cdot \mathbf{0.8}P(A and B)=0.5⋅0.8 P(A and B)=0.4P(A \text{ and } B) = \mathbf{0.4}P(A and B)=0.4So, the probability of both *A* and *B* happening is **0.4**.

STEP 7

P(A and B)=0.4P(A \text{ and } B) = 0.4P(A and B)=0.4

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Question< 13 14 15 $=16$ $\equiv 17$ 18 $=19$
Let $A$ and $B$ be events with $P(A)=0.8, P(B)=0.6$ , and $P(B \mid A)=0.5$ . Find $P(A$ and $B)$ . $P(A \text { and } B)=$ $\square$

1. Understand Conditional Probability
2. Calculate Joint Probability

Now, we just plug in the values we know: $P(A \text{ and } B) = \mathbf{0.5} \cdot \mathbf{0.8}$ $P(A \text{ and } B) = \mathbf{0.4}$ So, the probability of both A and B happening is 0.4.

$P(A \text{ and } B) = 0.4$