In a two-way table, “given that” signals conditional probability: you’re finding a probability within a smaller group. The condition chooses the denominator; the target chooses the numerator. Add the counts in the conditioned column, then find the cell where that column meets the requested row. Dividing by the original total answers a different question.
Hints
- Hint 1
The words given that restrict which cards are still possible before you calculate a probability. If the assigned table has an umbrella, which column of cards remains possible?
- Hint 2
The denominator is the total in the umbrella column, not the total number of cards. Add the umbrella counts from all three areas. Then find the cell that matches both Area C and umbrella.
Step-by-step
Restrict to the umbrella column
Step 1Keep only eligible cards
“Given that the assigned table has an umbrella” is a condition: it rules out every card in the No umbrella column. So only cards in the Umbrella column can count. Using all cards as the denominator would include cards the condition has ruled out.
- Step 2
Find the umbrella total
The Umbrella column has cards from Area A, from Area B, and from Area C. Type in Desmos; it shows . So is the denominator, the total number of eligible cards.
- Step 3
Count Area C within that total
The cell for Area C and Umbrella contains . That’s the numerator: the eligible cards that also match Area C. Type in Desmos; it shows about , and the fraction button displays . So the probability that the table is in Area C, given that it has an umbrella, is . Choice B.