3 Conditional Probability and Independence Free Online FlashCards

What is conditional probability?

For events $A$ and $B$ with $P(B)>0$, $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$.

Is conditional probability symmetric?

No. Generally, $P(A\mid B)\neq P(B\mid A)$ because the two conditional probabilities use different denominators.

State the multiplication rule for two events.

The multiplication rule is $P(A\cap B)=P(B)P(A\mid B)=P(A)P(B\mid A)$.

Two red balls without replacement: what is the probability?

$P(R_1\cap R_2)=\frac{5}{8}\cdot\frac{4}{7}=\frac{5}{14}$. The second factor changes because the first red ball is not replaced.

How does the multiplication rule extend to three events?

For three events, $P(A\cap B\cap C)=P(A)P(B\mid A)P(C\mid A\cap B)$.

What conditions define a partition?

A partition consists of pairwise disjoint events whose union is the entire sample space, with each event having a well-defined probability.

State the law of total probability.

If $B_1,\ldots,B_k$ partition the sample space, then $P(A)=\sum_{i=1}^{k}P(A\mid B_i)P(B_i)$.

What is the overall defect rate across the three factories?

The overall defect probability is $(0.02)(0.50)+(0.03)(0.30)+(0.05)(0.20)=0.029$, or $2.9\%$.

State Bayes’ theorem.

Bayes’ theorem is $P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$. It reverses the direction of conditioning.

Given a defect, what is the probability of Factory 3?

$P(F_3\mid D)=\frac{(0.05)(0.20)}{0.029}\approx0.3448$, so about $34.5\%$ of defective products came from Factory 3.

What is the main test for independence?

Events $A$ and $B$ are independent exactly when $P(A\cap B)=P(A)P(B)$.

What complement property follows from independence?

If $A$ and $B$ are independent, then $(A,B^c)$, $(A^c,B)$, and $(A^c,B^c)$ are also independent.