Practice Problems: Probability and Contingency Tables

Sample Population and Category Frequencies

• The sample consists of a categorical distribution of cat coat types and patterns comprising a total count of $97$ individuals:
  • Tabby: $33$
  • Tuxedo: $24$
  • Calico: $16$
  • Lynx Point: $10$
  • Spotted: $8$
  • Snowshoe: $6$
  • Total Count: $33 + 24 + 16 + 10 + 8 + 6 = 97$

Event Definitions and Marginal Probabilities

• Primary events are designated based on specific coat patterns within the population:
  • Event $A$ represents selecting a cat with a Tuxedo pattern.
  • Event $B$ represents selecting a cat with a Lynx Point pattern.
• The marginal probabilities of the events are calculated directly from their sample counts relative to the total population size of $97$:
  • $P(A) = \frac{24}{97} \approx 0.25$
  • $P(B) = \frac{10}{97} \approx 0.10$
• The complementary probabilities (the probability of an event not occurring) are determined using the complement rule $P(E') = 1 - P(E)$:
  • Complement of event $A$: $P(A') = 1 - 0.25 = 0.75$
  • Complement of event $B$: $P(B') = 1 - 0.10 = 0.90$

Two-Way Joint Probability Table

• A two-way probability grid is constructed to model joint events under the operational assumption of independence, where the joint probability of two events $X$ and $Y$ is evaluated via the multiplication rule: $P(X \cap Y) = P(X) \times P(Y)$.
• Joint probability calculations for each cell in the contingency layout:
  • Intersection of $A$ and $B$:
  • $P(A \cap B) = P(A) \times P(B)$
  • $P(A \cap B) = 0.25 \times 0.10 = 0.025 \approx 0.03$
  • Intersection of $A'$ and $B$:
  • $P(A' \cap B) = P(A') \times P(B)$
  • $P(A' \cap B) = 0.75 \times 0.10 = 0.075 \approx 0.08$
  • Intersection of $A$ and $B'$:
  • $P(A \cap B') = P(A) \times P(B')$
  • $P(A \cap B') = 0.25 \times 0.90 = 0.225 \approx 0.23$
  • Intersection of $A'$ and $B'$:
  • $P(A' \cap B') = P(A') \times P(B')$
  • $P(A' \cap B') = 0.75 \times 0.90 = 0.675 \approx 0.68$

Marginal Probability Reconstructions and Rounding Observations

• Row totals reconstruct the marginal probabilities for event $B$ and its complement $B'$ by summing the calculated joint probabilities:
  • Reconstructed $P(B) = P(A \cap B) + P(A' \cap B) = 0.03 + 0.08 = 0.11$
  • Reconstructed $P(B') = P(A \cap B') + P(A' \cap B') = 0.23 + 0.68 = 0.91$
  • Total probability across rows: $P(B) + P(B') = 0.11 + 0.91 = 1.02$
• Column totals reconstruct the marginal probabilities for event $A$ and its complement $A'$ by summing down each column:
  • Reconstructed $P(A) = P(A \cap B) + P(A \cap B') = 0.03 + 0.23 = 0.26$
  • Reconstructed $P(A') = P(A' \cap B) + P(A' \cap B') = 0.08 + 0.68 = 0.76$
  • Total probability across columns: $P(A) + P(A') = 0.26 + 0.76 = 1.02$
• Note on rounding discrepancy:
  • Intermediate rounding of each product to two decimal places introduces slight discrepancy between the original marginal probabilities ($P(A) = 0.25$, $P(B) = 0.10$) and the marginal sums reconstructed from rounded joint probabilities ($P(A) = 0.26$, $P(B) = 0.11$), producing total sums of $1.02$ rather than precisely $1.00$.