Applying the Addition Rule for Probability - Statistics and Probability

1. What is the equivalent notation statement for $P(A\text{ and }B)$ using the intersection symbol?

   $P(A\cap B)$. Intersection symbol represents outcomes in both events.

2. What is $P(A\text{ or }B)$ if events $A$ and $B$ are mutually exclusive?

   $P(A\text{ or }B)=P(A)+P(B)$. Mutually exclusive means no overlap, so $P(A\text{ and }B)=0$.

3. State the Addition Rule formula for $P(A\text{ or }B)$ in terms of $P(A)$, $P(B)$, and $P(A\text{ and }B)$.

   $P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$. Subtracts overlap to avoid counting shared outcomes twice.

4. Identify the error: A student used $P(A\text{ or }B)=P(A)+P(B)$ when $P(A\text{ and }B)\neq 0$.

   Missing subtraction of $P(A\text{ and }B)$. Must subtract overlap when events aren't disjoint.

5. Find $P(A\text{ or }B)$ if $P(A)=\frac{1}{4}$, $P(B)=\frac{1}{3}$, and $P(A\text{ and }B)=\frac{1}{12}$.

   $\frac{1}{2}$. $\frac{1}{4}+\frac{1}{3}-\frac{1}{12}=\frac{3+4-1}{12}=\frac{1}{2}$.

6. What value does the Addition Rule subtract to correct for double counting in $P(A\text{ or }B)$?

   $P(A\text{ and }B)$. Removes the overlap where both events occur together.

7. Find $P(A\text{ and }B)$ if $P(A)=0.55$, $P(B)=0.30$, and $P(A\text{ or }B)=0.70$.

   $0.15$. $0.55+0.30-0.70=0.15$ by rearranging.

8. What is the equivalent notation statement for $P(A\text{ or }B)$ using the union symbol?

   $P(A\cup B)$. Union symbol represents the set of outcomes in either event.

9. Identify the condition that makes $P(A \text{ and } B)=0$ in the Addition Rule.

   $A$ and $B$ are mutually exclusive (disjoint). Events can't happen simultaneously.

10. State the Addition Rule formula for $P(A \text{ or } B)$ in terms of $P(A)$, $P(B)$, and $P(A \text{ and } B)$.

    $P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$. Subtract overlap to avoid counting twice.

11. Find $P(A\text{ or }B)$ if $A$ and $B$ are mutually exclusive with $P(A)=0.18$ and $P(B)=0.22$.

    $0.40$. Mutually exclusive: just add $0.18+0.22=0.40$.

12. Which probability is subtracted in $P(A\text{ or }B)=P(A)+P(B)-\square$ to avoid double counting?

    $P(A\text{ and }B)$. Overlap is counted in both $P(A)$ and $P(B)$.

13. Find $P(A\text{ or }B)$ if $A\subseteq B$, $P(A)=0.20$, and $P(B)=0.60$.

    $0.60$. Since $A\subseteq B$, $A\cup B=B$.

14. Choose the correct statement: If $A\subseteq B$, what is $P(A\text{ or }B)$ equal to?

    $P(B)$. If $A$ is subset of $B$, union equals $B$.

15. Find $P(A\text{ or }B)$ given $P(A)=0.25$, $P(B)=0.50$, and $P(A\text{ and }B)=0.05$.

    $0.70$. Apply: $0.25+0.50-0.05=0.70$.

16. Find $P(A\text{ and }B)$ if $P(A)=0.48$, $P(B)=0.52$, and $P(A\text{ or }B)=0.80$.

    $0.20$. $0.48+0.52-0.80=0.20$ by rearranging.

17. Find $P(B)$ given $P(A)=0.35$, $P(A\text{ and }B)=0.15$, and $P(A\text{ or }B)=0.65$.

    $0.45$. Rearrange: $P(B)=0.65-0.35+0.15=0.45$.

18. Find $P(A\text{ or }B)$ given $P(A)=0.40$, $P(B)=0.30$, and $P(A\text{ and }B)=0.10$.

    $0.60$. Apply: $0.40+0.30-0.10=0.60$.

19. Find $P(A\text{ or }B)$ if $P(A)=0.40$, $P(B)=0.35$, and $P(A\text{ and }B)=0.10$.

    $0.65$. $0.40+0.35-0.10=0.65$ by Addition Rule.

20. Find $P(A\text{ and }B)$ if $A$ and $B$ are independent with $P(A)=0.20$ and $P(B)=0.50$.

    $0.10$. Independent events: multiply $0.20(0.50)=0.10$.

21. What is the meaning of $P(A \text{ or } B)$ in words (inclusive or)?

    Probability that at least one of $A$ or $B$ occurs. Union includes either or both events.

22. What is the formula for $P(A \text{ or } B)$ when $A$ and $B$ are mutually exclusive?

    $P(A\text{ or }B)=P(A)+P(B)$. No overlap when events can't occur together.

23. Find $P(A\text{ or }B)$ given counts: $|A|=30$, $|B|=25$, $|A\cap B|=10$, total $N=100$.

    $0.45$. Count unique: $(30+25-10)/100=45/100=0.45$.

24. Find $P(A\text{ and }B)$ given $P(A)=0.60$, $P(B)=0.50$, and $P(A\text{ or }B)=0.80$.

    $0.30$. Rearrange: $P(A\text{ and }B)=0.60+0.50-0.80=0.30$.

25. Find $P(A)$ given $P(B)=0.55$, $P(A\text{ and }B)=0.20$, and $P(A\text{ or }B)=0.75$.

    $0.40$. Rearrange: $P(A)=0.75-0.55+0.20=0.40$.

26. Find $P(A\text{ and }B)$ if $P(A)=\frac{3}{5}$, $P(B)=\frac{1}{2}$, and $P(A\text{ or }B)=\frac{4}{5}$.

    $\frac{3}{10}$. $\frac{3}{5}+\frac{1}{2}-\frac{4}{5}=\frac{6+5-8}{10}=\frac{3}{10}$.

27. Interpretation check: If $P(A\text{ or }B)=0.72$, what does $0.72$ represent in context?

    Chance that at least one of $A$ or $B$ occurs. Probability that event $A$ or event $B$ happens.

28. What condition on probabilities indicates that $A$ and $B$ are mutually exclusive?

    $P(A\text{ and }B)=0$. Events cannot occur together when mutually exclusive.

29. Find $P(A\cup B)$ from a table where $P(A)=0.52$, $P(B)=0.27$, and $P(A\cap B)=0.12$.

    $0.67$. Apply: $0.52+0.27-0.12=0.67$.

30. What is the complement form for $P(A\text{ or }B)$ using $P(A^c\text{ and }B^c)$?

    $P(A\text{ or }B)=1-P(A^c\text{ and }B^c)$. By De Morgan's law: at least one occurs = not both fail.

31. What is the meaning of $P(A \text{ and } B)$ in words?

    Probability that both $A$ and $B$ occur. The intersection of events $A$ and $B$.

32. Find $P(A\text{ or }B)$ if $P(A)=0.20$, $P(B)=0.50$, and $A$ and $B$ are mutually exclusive.

    $0.70$. $0.20+0.50-0=0.70$ since mutually exclusive.

33. Find $P(A\text{ or }B)$ if $A$ and $B$ are independent with $P(A)=0.30$ and $P(B)=0.40$.

    $0.58$. Apply: $0.30+0.40-0.30(0.40)=0.58$.

34. What is the formula for $P(A \text{ and } B)$ rewritten from the Addition Rule?

    $P(A\text{ and }B)=P(A)+P(B)-P(A\text{ or }B)$. Rearrange the Addition Rule by solving for $P(A\text{ and }B)$.

35. Find $P(A\text{ or }B)$ if $P(A)=0.15$, $P(B)=0.10$, and $P(A\text{ and }B)=0$.

    $0.25$. $0.15+0.10-0=0.25$ for disjoint events.

36. Identify the correct expression for $P(A\text{ or }B)$ when $A$ and $B$ are independent.

    $P(A\text{ or }B)=P(A)+P(B)-P(A)P(B)$. For independent events, $P(A\text{ and }B)=P(A)P(B)$.

37. Find and correct the error: $P(A\text{ or }B)=P(A)+P(B)+P(A\text{ and }B)$.

    Correct: $P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$. Should subtract, not add, the intersection term.

38. Find $P(A\text{ or }B)$ if $P(A)=0.90$, $P(B)=0.30$, and $P(A\text{ and }B)=0.25$.

    $0.95$. $0.90+0.30-0.25=0.95$ by Addition Rule.

39. Find $P(A\text{ or }B)$ from counts: $n(A)=25$, $n(B)=30$, $n(A\cap B)=10$, total $n=60$.

    $\frac{3}{4}$. $\frac{25+30-10}{60}=\frac{45}{60}=\frac{3}{4}$.

40. Which inequality must always be true for valid events: $P(A\text{ or }B)\leq \ ?$

    $P(A\text{ or }B)\leq 1$. Probability cannot exceed 1 (certainty).

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