Conditional Probability and Independence - Statistics and Probability

1. State an equivalent independence condition using only $P(A\cap B)$, $P(A)$, and $P(B)$.

   $A$ and $B$ are independent if $P(A\cap B)=P(A)P(B)$. Product rule holds when events don't influence each other.

2. Decide whether $A$ and $B$ are independent: $P(A)=0.5$, $P(B)=0.4$, $P(A\cap B)=0.2$.

   Independent, since $P(A\cap B)=P(A)P(B)=0.2$. Checks if $P(A\cap B)=P(A)P(B)$: $0.2=0.5\times 0.4$ ✓.

3. State the multiplication rule that expresses $P(A\cap B)$ using $P(A\mid B)$ and $P(B)$.

   $P(A\cap B)=P(A\mid B)P(B)$. Rearranges the conditional probability formula.

4. Find $P(B\mid A)$ if $A$ and $B$ are independent and $P(B)=0.15$.

   $P(B\mid A)=0.15$. Independence means $P(B\mid A)=P(B)$.

5. Identify the error: A student writes $P(A\mid B)=\frac{P(A\cup B)}{P(B)}$. What is the correction?

   Use $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Union $\cup$ should be intersection $\cap$.

6. What does the notation $P(A\mid B)$ mean in words?

   Probability that $A$ occurs given that $B$ has occurred. The vertical bar means "given" or "assuming."

7. State the independence rule that connects $P(A\cap B)$, $P(A)$, and $P(B)$.

   If independent, $P(A\cap B)=P(A)P(B)$. For independent events, joint probability equals product of individual probabilities.

8. Find $P(B)$ given $P(A\cap B)=0.18$ and $P(A\mid B)=0.30$.

   $P(B)=0.6$. Rearranges $P(A\cap B)=P(A\mid B)P(B)$ to solve for $P(B)=\frac{0.18}{0.30}$.

9. Decide whether $A$ and $B$ are independent if $P(A)=0.40$ and $P(A\mid B)=0.25$.

   Not independent. Since $0.25\neq 0.40$, conditioning changes probability.

10. Find $P(A\cap B)$ given $P(A\mid B)=0.25$ and $P(B)=0.60$.

    $P(A\cap B)=0.15$. Uses multiplication rule: $P(A\cap B)=P(A\mid B)P(B)=0.25\times 0.60$.

11. Find $P(A\cap B)$ if $P(A\mid B)=0.25$ and $P(B)=0.60$.

    $0.15$. Multiply: $P(A\cap B)=0.25\times 0.60=0.15$.

12. What is the definition of independence using conditional probability $P(B\mid A)$?

    If $P(A)>0$, independence means $P(B\mid A)=P(B)$. Independence means knowing $A$ doesn't change the probability of $B$.

13. What is the definition of independence using conditional probability $P(A\mid B)$?

    If $P(B)>0$, independence means $P(A\mid B)=P(A)$. Independence means knowing $B$ doesn't change the probability of $A$.

14. State the multiplication rule that expresses $P(A\cap B)$ using $P(B\mid A)$ and $P(A)$.

    $P(A\cap B)=P(B\mid A)P(A)$. Alternative form using reversed conditioning.

15. Which option is equivalent to independence: $P(A\mid B)=P(A)$ or $P(A\mid B)=P(B)$?

    $P(A\mid B)=P(A)$. Independence means conditioning doesn't change probability, not that they're equal.

16. Identify the correct relationship when $A$ and $B$ are independent and $P(B)>0$.

    $P(A\cap B)=P(A\mid B)P(B)=P(A)P(B)$. Substitutes $P(A)$ for $P(A\mid B)$ when events are independent.

17. State the formula for conditional probability $P(A\mid B)$ in terms of $P(A\cap B)$ and $P(B)$.

    $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$, with $P(B)>0$. Divides joint probability by the condition's probability.

18. What condition must be true for $P(A\mid B)$ to be defined?

    $P(B)>0$. Can't condition on an impossible event.

19. Find $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)=0.35$.

    $P(A\mid B)=0.35$. For independent events, $P(A\mid B)=P(A)$ regardless of $B$.

20. State the independence condition using conditional probability of $A$ given $B$.

    $A$ and $B$ are independent if $P(A\mid B)=P(A)$. Knowing $B$ doesn't change $A$'s probability.

21. Find $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)=0.72$.

    $P(A\mid B)=0.72$. Independence means $P(A\mid B)=P(A)$.

22. Find $P(A\mid B)$ given $P(A\cap B)=0.12$ and $P(B)=0.30$.

    $P(A\mid B)=0.4$. Uses $P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{0.12}{0.30}$.

23. Find $P(A\mid B)$ if $P(A\cap B)=0.12$ and $P(B)=0.30$.

    $0.4$. Apply $P(A\mid B)=\frac{0.12}{0.30}=0.4$.

24. Find $P(B)$ if $P(A\cap B)=0.09$ and $P(A\mid B)=0.30$.

    $0.3$. Rearrange: $P(B)=\frac{0.09}{0.30}=0.3$.

25. Find $P(B\mid A)$ if $A$ and $B$ are independent and $P(B)=0.80$.

    $P(B\mid A)=0.80$. For independent events, $P(B\mid A)=P(B)$ regardless of $A$.

26. Identify the value of $P(A\cap B)$ from $P(A)=0.3$, $P(B)=0.5$, assuming independence.

    $P(A\cap B)=0.15$. Uses independence formula: $P(A\cap B)=P(A)P(B)=0.3\times 0.5$.

27. Compute $P(A\mid B)$ from the table: $A\cap B=15$, $A^c\cap B=35$.

    $P(A\mid B)=\frac{15}{50}=0.3$. Total in $B$ is $15+35=50$, so $P(A\mid B)=\frac{15}{50}$.

28. Identify the correct expression for $P(A\cap B)$ in terms of $P(A)$ and $P(B\mid A)$.

    $P(A\cap B)=P(A)P(B\mid A)$. Applies the multiplication rule correctly.

29. Find $P(A\cap B)$ if $A$ and $B$ are independent, $P(A)=0.30$, and $P(B)=0.40$.

    $0.12$. For independent events, multiply probabilities.

30. Which probability is updated by new information: $P(A)$ or $P(A\mid B)$?

    $P(A\mid B)$. Conditioning incorporates new information.

31. Decide whether $A$ and $B$ are independent if $P(A)=0.50$ and $P(A\mid B)=0.50$.

    Independent. Since $P(A\mid B)=P(A)$, they're independent.

32. State the independence condition using conditional probability of $B$ given $A$.

    $A$ and $B$ are independent if $P(B\mid A)=P(B)$. Knowing $A$ doesn't change $B$'s probability.

33. Decide whether $A$ and $B$ are independent: $P(A)=0.6$, $P(B)=0.5$, $P(A\cap B)=0.25$.

    Not independent, since $P(A)P(B)=0.3\ne 0.25$. Checks if $P(A\cap B)=P(A)P(B)$: $0.25\ne 0.6\times 0.5=0.3$ ✗.

34. Find $P(A)$ assuming independence, given $P(A\cap B)=0.14$ and $P(B)=0.70$.

    $P(A)=0.2$. Uses independence: $P(A)=\frac{P(A\cap B)}{P(B)}=\frac{0.14}{0.70}$.

35. Decide whether $A$ and $B$ are independent if $P(A\cap B)=0.10$, $P(A)=0.20$, and $P(B)=0.50$.

    Independent. Check: $0.10=0.20\times 0.50$ confirms independence.

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