Let $$ A $$ and $$ B $$ be two dependent events. If $$ P(A)=0.5, P(B)= $$ 0.75 and $$ P(A \cap B)=0.4 $$. What is the value of $$ P(A / B) $$ ? $$ \begin{array}{ll} ext { a. equal to } P(B) & ext { b. less than } P(A \cap B) \ ext { c. less than } P(B / A) & ext { d. equal to } P(B / A)\end{array} $$

Question

Let $$ A $$ and $$ B $$ be two dependent events. If $$ P(A)=0.5, P(B)= $$ 0.75 and $$ P(A \cap B)=0.4 $$. What is the value of $$ P(A / B) $$ ? $$ \begin{array}{ll}	ext { a. equal to } P(B) & 	ext { b. less than } P(A \cap B) \ 	ext { c. less than } P(B / A) & 	ext { d. equal to } P(B / A)\end{array} $$

UpStudy AI · Accepted Answer

We are given:
$$
P(A)=0.5,\quad P(B)=0.75,\quad P(A \cap B)=0.4.
$$

The conditional probability $$ P(A \mid B) $$ is defined as:
$$
P(A \mid B)=\frac{P(A \cap B)}{P(B)}.
$$

Substitute the given values:
$$
P(A \mid B)=\frac{0.4}{0.75}\approx 0.533.
$$

Now, let’s compare this with the provided options:
- Option a: $$ P(B)=0.75 $$. This is not equal to $$ 0.533 $$.
- Option b: $$ P(A \cap B)=0.4 $$. $$ 0.533 $$ is not less than $$ 0.4 $$.
- Option c: $$ P(B \mid A)=\frac{P(A \cap B)}{P(A)}=\frac{0.4}{0.5}=0.8 $$. Here, $$ P(A \mid B) \approx 0.533 $$ is less than $$ 0.8 $$.
- Option d: $$ P(B \mid A)=0.8 $$. This is not equal to $$ 0.533 $$.

Thus, the correct option is that $$ P(A \mid B) $$ is less than $$ P(B \mid A) $$.
$$ P(A \mid B) $$ is less than $$ P(B \mid A) $$.

Let and be two dependent events. If
0.75 and . What is the value of ?

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Let and be two dependent events. If 0.75 and . What is the value of ?