students in mrs. barness class determined the probability that she will check homework on a randomly chosen day is 0.42. they also determined the probability that she will give a pop quiz when she checks homework is 0.6, and the probability that she will give a pop quiz when she does not check homework is 0.9. the probabilities are displayed in the tree diagram. what is the probability that mrs. barnes checks homework if the students take a pop quiz? 0.25 0.33 0.67 0.77

Question

Accepted Answer

# Explanation:
## Step1: Use the formula for conditional probability
The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Let $A$ be the event that Mrs. Barnes checks homework and $B$ be the event that there is a pop - quiz.

The probability that Mrs. Barnes checks homework and gives a pop - quiz is $P(A\cap B)=0.42\times0.6$.

The probability that there is a pop - quiz is $P(B)=0.42\times0.6 + 0.58\times0.9$.

## Step2: Calculate $P(A\cap B)$
$$
\begin{align*}
P(A\cap B)&=0.42\times0.6\
&=0.252
\end{align*}
$$

## Step3: Calculate $P(B)$
$$
\begin{align*}
P(B)&=0.42\times0.6+0.58\times0.9\
&=0.252 + 0.522\
&=0.774
\end{align*}
$$

## Step4: Calculate $P(A|B)$
$$
\begin{align*}
P(A|B)&=\frac{P(A\cap B)}{P(B)}\
&=\frac{0.252}{0.774}\
&\approx0.33
\end{align*}
$$

# Answer:
0.33

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QUESTION IMAGE

Question

Explanation:

Step1: Use the formula for conditional probability

Step2: Calculate \(P(A\cap B)\)

Step3: Calculate \(P(B)\)

Step4: Calculate \(P(A|B)\)

Answer: