Using Two-Way Tables for Probability - Statistics and Probability

1. What is the definition of a two-way frequency table?

   A table of counts for data classified by two categorical variables. Shows frequencies for combinations of two categorical variables.

2. Find $P(A\cap B)$ if the cell count for $(A,B)$ is $18$ out of a grand total of $120$.

   $\frac{18}{120}=0.15$. Apply joint probability formula: cell count over grand total.

3. Decide if $A$ and $B$ are independent when $P(A)=0.4$, $P(B)=0.5$, $P(A\cap B)=0.2$.

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

4. What equation must hold for events $A$ and $B$ to be independent using probabilities?

   $P(A\cap B)=P(A)P(B)$. Independence means joint probability equals product of marginals.

5. Find $P(B\mid A)$ if the count in $(A,B)$ is $9$ and the total for $A$ is $36$.

   $\frac{9}{36}=0.25$. Apply conditional formula with $A$ as the condition.

6. Find $P(B\mid A)$ if $\text{count}(A\cap B)=9$, $\text{count}(A)=30$.

   $\frac{9}{30}=0.30$. Divide joint count by condition's marginal.

7. What conditional-probability equality indicates $A$ and $B$ are independent (when defined)?

   $P(A\mid B)=P(A)$. Independence means conditioning doesn't change the probability.

8. What is the complement rule for a conditional probability $P(A\mid B)$?

   $P(A^c\mid B)=1-P(A\mid B)$. Complement probabilities sum to 1 within the conditional space.

9. Find $P(A\mid B)$ if $\text{count}(A\cap B)=12$ and $\text{count}(B)=40$.

   $\frac{12}{40}=0.30$. Apply conditional probability formula.

10. Find $P(A)$ if the marginal total for $A$ is $45$ out of a grand total of $150$.

    $\frac{45}{150}=0.30$. Apply marginal probability formula.

11. What is the formula for the probability of an event using a marginal total?

    $P(A)=\frac{\text{marginal total for }A}{\text{grand total}}$. Divide row/column sum by total count.

12. Identify the sample space when using a two-way table to model outcomes.

    All table cells (each category pair) plus the grand total as the denominator. Each cell represents an outcome; grand total normalizes probabilities.

13. What is a row total in a two-way frequency table?

    The sum of counts across a single row. Adds all values in that row to find the row's marginal total.

14. Compute $P(A^c\mid B)$ if $P(A\mid B)=0.72$.

    $1-0.72=0.28$. Use complement rule: probabilities sum to 1.

15. What is the difference between joint and conditional probability in a two-way table?

    Joint uses grand total; conditional uses the given category total. Joint divides by all outcomes; conditional divides by given category.

16. What formula gives the conditional probability $P(A\mid B)$ from a two-way table?

    $P(A\mid B)=\frac{\text{count in }A\text{ and }B}{\text{total count in }B}$. Restricts to condition $B$ by dividing by $B$'s total.

17. Find $P(A\mid B)$ if the count in $(A,B)$ is $7$ and the total for $B$ is $20$.

    $\frac{7}{20}=0.35$. Apply conditional formula: cell count over condition's total.

18. Find $P(A\mid B)$ if $P(A\cap B)=0.18$ and $P(B)=0.6$.

    $\frac{0.18}{0.6}=0.30$. Apply $P(A|B) = \frac{P(A \cap B)}{P(B)}$.

19. What formula gives the joint probability $P(A \cap B)$ from a two-way table?

    $P(A \cap B)=\frac{\text{count in }A\text{ and }B}{\text{grand total}}$. Divides the cell count by grand total for joint probability.

20. What is a joint frequency in a two-way table?

    A count in an interior cell for a specific pair of categories. Frequency where two specific categories intersect.

21. Decide if $A$ and $B$ are independent if $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$.

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

22. What formula gives the marginal probability $P(A)$ from a two-way table?

    $P(A)=\frac{\text{row or column total for }A}{\text{grand total}}$. Divides marginal total by grand total for single-event probability.

23. What condition using probabilities shows events $A$ and $B$ are independent?

    $P(A\cap B)=P(A)P(B)$. Independent events satisfy the multiplication rule.

24. Decide if $A$ and $B$ are independent when $P(A)=0.3$, $P(B)=0.6$, $P(A\cap B)=0.25$.

    Not independent, since $0.3\times 0.6=0.18\ne 0.25$. Product $0.18$ doesn't equal joint probability $0.25$.

25. Which equality using conditional probability shows $A$ and $B$ are independent?

    $P(A\mid B)=P(A)$. Conditioning doesn't change probability when independent.

26. What is a two-way frequency table used for when each object has two categorical variables?

    A table of counts for all combinations of two categorical variables. Organizes counts when objects have two categorical attributes.

27. What is the grand total in a two-way frequency table?

    The sum of all cell counts in the table. Add all interior cells or all marginal totals.

28. Identify the correct comparison to check independence in a table: compare $P(A\mid B)$ to what value?

    Compare $P(A\mid B)$ to $P(A)$. Equal values indicate independence.

29. Find $P(A)$ if the row total for $A$ is $54$ out of a grand total of $90$.

    $\frac{54}{90}=0.6$. Marginal probability uses row total over grand total.

30. What is a marginal total in a two-way table?

    A row or column total showing counts for one variable alone. Sum of all values in that row or column.

31. What is the formula for conditional probability from a two-way table?

    $P(A\mid B)=\frac{\text{count}(A\cap B)}{\text{count}(B)}$. Divide joint count by condition's total count.

32. Choose the correct interpretation of $P(\text{Science}\mid \text{Grade }10)$ from a two-way table.

    Probability a student prefers Science among only Grade $10$ students. Conditional restricts to Grade 10 subset only.

33. What is the formula for joint probability using a two-way table cell count?

    $P(A \cap B)=\frac{\text{cell count}}{\text{grand total}}$. Divide the cell count by total count for joint probability.

34. What is a column total in a two-way frequency table?

    The sum of counts down a single column. Adds all values in that column to find the column's marginal total.

35. Decide if $A$ and $B$ are independent if $P(A)=0.3$, $P(B)=0.6$, and $P(A\cap B)=0.25$.

    Not independent, since $0.25\ne 0.3\times 0.6$. Check if $P(A \cap B) = P(A)P(B)$: $0.25 \ne 0.18$ ✗

36. Find $P(A\cap B)$ if the cell count for $(A,B)$ is $18$ and the grand total is $120$.

    $\frac{18}{120}=0.15$. Apply joint probability formula: cell/total.

37. Identify the sample space when a two-way frequency table is used for probability.

    All individuals in the table; total outcomes equal the grand total. Each person counted is one possible outcome.

38. Find $P(A\mid B)$ if the count in $(A,B)$ is $12$ and the total for $B$ is $48$.

    $\frac{12}{48}=0.25$. Apply conditional formula: cell count over condition's total.

39. Compute the missing cell: grand total $200$, other three cells in a $2\times 2$ table sum to $155$.

    $200-155=45$. All cells must sum to grand total.

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