Using Two-Way Tables for Probability - Statistics
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What is a joint frequency in a two-way table?
What is a joint frequency in a two-way table?
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A count in an interior cell for a specific pair of categories. Frequency where two specific categories intersect.
A count in an interior cell for a specific pair of categories. Frequency where two specific categories intersect.
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What is the definition of a two-way frequency table?
What is the definition of a two-way frequency table?
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A table of counts for data classified by two categorical variables. Shows frequencies for combinations of two categorical variables.
A table of counts for data classified by two categorical variables. Shows frequencies for combinations of two categorical variables.
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Identify the correct comparison to check independence in a table: compare $P(A\mid B)$ to what value?
Identify the correct comparison to check independence in a table: compare $P(A\mid B)$ to what value?
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Compare $P(A\mid B)$ to $P(A)$. Equal values indicate independence.
Compare $P(A\mid B)$ to $P(A)$. Equal values indicate independence.
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Choose the correct interpretation of $P(\text{Science}\mid \text{Grade }10)$ from a two-way table.
Choose the correct interpretation of $P(\text{Science}\mid \text{Grade }10)$ from a two-way table.
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Probability a student prefers Science among only Grade $10$ students. Conditional restricts to Grade 10 subset only.
Probability a student prefers Science among only Grade $10$ students. Conditional restricts to Grade 10 subset only.
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Compute the missing cell: grand total $200$, other three cells in a $2\times 2$ table sum to $155$.
Compute the missing cell: grand total $200$, other three cells in a $2\times 2$ table sum to $155$.
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$200-155=45$. All cells must sum to grand total.
$200-155=45$. All cells must sum to grand total.
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Find $P(B\mid A)$ if $\text{count}(A\cap B)=9$, $\text{count}(A)=30$.
Find $P(B\mid A)$ if $\text{count}(A\cap B)=9$, $\text{count}(A)=30$.
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$\frac{9}{30}=0.30$. Divide joint count by condition's marginal.
$\frac{9}{30}=0.30$. Divide joint count by condition's marginal.
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Find $P(A\mid B)$ if $P(A\cap B)=0.18$ and $P(B)=0.6$.
Find $P(A\mid B)$ if $P(A\cap B)=0.18$ and $P(B)=0.6$.
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$\frac{0.18}{0.6}=0.30$. Apply $P(A|B) = \frac{P(A \cap B)}{P(B)}$.
$\frac{0.18}{0.6}=0.30$. Apply $P(A|B) = \frac{P(A \cap B)}{P(B)}$.
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Decide if $A$ and $B$ are independent when $P(A)=0.3$, $P(B)=0.6$, $P(A\cap B)=0.25$.
Decide if $A$ and $B$ are independent when $P(A)=0.3$, $P(B)=0.6$, $P(A\cap B)=0.25$.
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Not independent, since $0.3\times 0.6=0.18\ne 0.25$. Product $0.18$ doesn't equal joint probability $0.25$.
Not independent, since $0.3\times 0.6=0.18\ne 0.25$. Product $0.18$ doesn't equal joint probability $0.25$.
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Decide if $A$ and $B$ are independent when $P(A)=0.4$, $P(B)=0.5$, $P(A\cap B)=0.2$.
Decide if $A$ and $B$ are independent when $P(A)=0.4$, $P(B)=0.5$, $P(A\cap B)=0.2$.
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Independent, since $0.4\times 0.5=0.2$. Check if $P(A)P(B) = P(A \cap B)$.
Independent, since $0.4\times 0.5=0.2$. Check if $P(A)P(B) = P(A \cap B)$.
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Find $P(A\mid B)$ if $\text{count}(A\cap B)=12$ and $\text{count}(B)=40$.
Find $P(A\mid B)$ if $\text{count}(A\cap B)=12$ and $\text{count}(B)=40$.
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$\frac{12}{40}=0.30$. Apply conditional probability formula.
$\frac{12}{40}=0.30$. Apply conditional probability formula.
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Find $P(A)$ if the marginal total for $A$ is $45$ out of a grand total of $150$.
Find $P(A)$ if the marginal total for $A$ is $45$ out of a grand total of $150$.
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$\frac{45}{150}=0.30$. Apply marginal probability formula.
$\frac{45}{150}=0.30$. Apply marginal probability formula.
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Find $P(A\cap B)$ if the cell count for $(A,B)$ is $18$ and the grand total is $120$.
Find $P(A\cap B)$ if the cell count for $(A,B)$ is $18$ and the grand total is $120$.
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$\frac{18}{120}=0.15$. Apply joint probability formula: cell/total.
$\frac{18}{120}=0.15$. Apply joint probability formula: cell/total.
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What is the grand total in a two-way frequency table?
What is the grand total in a two-way frequency table?
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The sum of all cell counts in the table. Add all interior cells or all marginal totals.
The sum of all cell counts in the table. Add all interior cells or all marginal totals.
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Identify the sample space when a two-way frequency table is used for probability.
Identify the sample space when a two-way frequency table is used for probability.
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All individuals in the table; total outcomes equal the grand total. Each person counted is one possible outcome.
All individuals in the table; total outcomes equal the grand total. Each person counted is one possible outcome.
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Which equality using conditional probability shows $A$ and $B$ are independent?
Which equality using conditional probability shows $A$ and $B$ are independent?
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$P(A\mid B)=P(A)$. Conditioning doesn't change probability when independent.
$P(A\mid B)=P(A)$. Conditioning doesn't change probability when independent.
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What condition using probabilities shows events $A$ and $B$ are independent?
What condition using probabilities shows events $A$ and $B$ are independent?
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$P(A\cap B)=P(A)P(B)$. Independent events satisfy the multiplication rule.
$P(A\cap B)=P(A)P(B)$. Independent events satisfy the multiplication rule.
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What is the formula for the probability of an event using a marginal total?
What is the formula for the probability of an event using a marginal total?
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$P(A)=\frac{\text{marginal total for }A}{\text{grand total}}$. Divide row/column sum by total count.
$P(A)=\frac{\text{marginal total for }A}{\text{grand total}}$. Divide row/column sum by total count.
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What is the formula for conditional probability from a two-way table?
What is the formula for conditional probability from a two-way table?
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$P(A\mid B)=\frac{\text{count}(A\cap B)}{\text{count}(B)}$. Divide joint count by condition's total count.
$P(A\mid B)=\frac{\text{count}(A\cap B)}{\text{count}(B)}$. Divide joint count by condition's total count.
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What is the formula for joint probability using a two-way table cell count?
What is the formula for joint probability using a two-way table cell count?
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$P(A \cap B)=\frac{\text{cell count}}{\text{grand total}}$. Divide the cell count by total count for joint probability.
$P(A \cap B)=\frac{\text{cell count}}{\text{grand total}}$. Divide the cell count by total count for joint probability.
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What is a marginal total in a two-way table?
What is a marginal total in a two-way table?
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A row or column total showing counts for one variable alone. Sum of all values in that row or column.
A row or column total showing counts for one variable alone. Sum of all values in that row or column.
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What formula gives the joint probability $P(A \cap B)$ from a two-way table?
What formula gives the joint probability $P(A \cap B)$ from a two-way table?
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$P(A \cap B)=\frac{\text{count in }A\text{ and }B}{\text{grand total}}$. Divides the cell count by grand total for joint probability.
$P(A \cap B)=\frac{\text{count in }A\text{ and }B}{\text{grand total}}$. Divides the cell count by grand total for joint probability.
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What is a column total in a two-way frequency table?
What is a column total in a two-way frequency table?
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The sum of counts down a single column. Adds all values in that column to find the column's marginal total.
The sum of counts down a single column. Adds all values in that column to find the column's marginal total.
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What is a two-way frequency table used for when each object has two categorical variables?
What is a two-way frequency table used for when each object has two categorical variables?
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A table of counts for all combinations of two categorical variables. Organizes counts when objects have two categorical attributes.
A table of counts for all combinations of two categorical variables. Organizes counts when objects have two categorical attributes.
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What is a row total in a two-way frequency table?
What is a row total in a two-way frequency table?
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The sum of counts across a single row. Adds all values in that row to find the row's marginal total.
The sum of counts across a single row. Adds all values in that row to find the row's marginal total.
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Identify the sample space when using a two-way table to model outcomes.
Identify the sample space when using a two-way table to model outcomes.
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All table cells (each category pair) plus the grand total as the denominator. Each cell represents an outcome; grand total normalizes probabilities.
All table cells (each category pair) plus the grand total as the denominator. Each cell represents an outcome; grand total normalizes probabilities.
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What is the difference between joint and conditional probability in a two-way table?
What is the difference between joint and conditional probability in a two-way table?
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Joint uses grand total; conditional uses the given category total. Joint divides by all outcomes; conditional divides by given category.
Joint uses grand total; conditional uses the given category total. Joint divides by all outcomes; conditional divides by given category.
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Find $P(A\cap B)$ if the cell count for $(A,B)$ is $18$ out of a grand total of $120$.
Find $P(A\cap B)$ if the cell count for $(A,B)$ is $18$ out of a grand total of $120$.
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$\frac{18}{120}=0.15$. Apply joint probability formula: cell count over grand total.
$\frac{18}{120}=0.15$. Apply joint probability formula: cell count over grand total.
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Find $P(A\mid B)$ if the count in $(A,B)$ is $12$ and the total for $B$ is $48$.
Find $P(A\mid B)$ if the count in $(A,B)$ is $12$ and the total for $B$ is $48$.
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$\frac{12}{48}=0.25$. Apply conditional formula: cell count over condition's total.
$\frac{12}{48}=0.25$. Apply conditional formula: cell count over condition's total.
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Find $P(B\mid A)$ if the count in $(A,B)$ is $9$ and the total for $A$ is $36$.
Find $P(B\mid A)$ if the count in $(A,B)$ is $9$ and the total for $A$ is $36$.
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$\frac{9}{36}=0.25$. Apply conditional formula with $A$ as the condition.
$\frac{9}{36}=0.25$. Apply conditional formula with $A$ as the condition.
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Decide if $A$ and $B$ are independent if $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$.
Decide if $A$ and $B$ are independent if $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$.
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Independent, since $0.2=0.4\times 0.5$. Check if $P(A \cap B) = P(A)P(B)$: $0.2 = 0.4 \times 0.5$ ✓
Independent, since $0.2=0.4\times 0.5$. Check if $P(A \cap B) = P(A)P(B)$: $0.2 = 0.4 \times 0.5$ ✓
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