Question

analyzing work caitlyn calculated the probability of the complement of rolling a number greater than 2 on a 6 - side number cube. she made her calculation as follows. p(less than or equal to 2): \\(\frac{\text{numbers greater than 2}}{\text{numbers less than or equal to 2}} = \frac{4}{2}\\) did she make an error? explain.

Explanation:

Step1: Recall probability formula

Probability of an event \( E \) is \( P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \). For the complement of an event (let \( A \) be "rolling greater than 2", complement \( A' \) is "rolling less than or equal to 2"), \( P(A') = 1 - P(A) \) or \( P(A')=\frac{\text{Number of outcomes in } A'}{\text{Total outcomes}} \).

Step2: Identify outcomes

A 6 - sided cube has numbers 1, 2, 3, 4, 5, 6.

• Outcomes for "greater than 2": 3, 4, 5, 6 (so 4 outcomes).
• Outcomes for "less than or equal to 2": 1, 2 (so 2 outcomes).
• Total outcomes: 6.

Step3: Analyze Caitlyn's error

Caitlyn used \( \frac{\text{Numbers greater than 2}}{\text{Numbers less than or equal to 2}} \), but probability should be based on total outcomes. Correctly, \( P(\text{less than or equal to 2})=\frac{\text{Outcomes for less than or equal to 2}}{\text{Total outcomes}}=\frac{2}{6}=\frac{1}{3} \), or using complement: \( P(\text{greater than 2})=\frac{4}{6}=\frac{2}{3} \), so \( P(\text{less than or equal to 2}) = 1-\frac{2}{3}=\frac{1}{3} \). Her formula is wrong as she divided by number of complement outcomes, not total.

Answer:

Yes, Caitlyn made an error. To find the probability of the complement (rolling less than or equal to 2), she should divide the number of outcomes for "less than or equal to 2" (which is 2: 1, 2) by the total number of possible outcomes (6, since the cube has 6 sides), not by the number of outcomes for "greater than 2". The correct probability is \( \frac{2}{6}=\frac{1}{3} \), but her calculation used an incorrect formula (dividing number of "greater than 2" outcomes by "less than or equal to 2" outcomes) leading to a wrong result.