12 Calculating Probabilities
Mathematics
Probability Theory
Statistical Reasoning
Problem Solving
Quantitative Analysis
12.1 Description:
This task involves calculating probabilities for various scenarios, including simple and compound events. It evaluates the model’s ability to apply probability theory, use appropriate formulas, and reason about likelihood in different contexts.
12.2 Modality:
Text only
12.3 Examples:
12.3.1 Example 1:
Input:
A standard deck of 52 cards contains 4 aces. If you draw two cards without replacement, what is the probability of drawing two aces?Output:
The probability of drawing two aces is 1/221 or approximately 0.45%.
Explanation:
1. Probability of drawing first ace: 4/52 = 1/13
2. Probability of drawing second ace, given first was an ace: 3/51
3. Probability of both events occurring: (1/13) * (3/51) = 3/663 = 1/221
4. 1/221 ≈ 0.0045 or 0.45%12.3.2 Example 2:
Input:
In a group of 100 people, 60 like chocolate, 50 like vanilla, and 30 like both flavors. What is the probability that a randomly selected person likes either chocolate or vanilla (or both)?Output:
The probability is 80% or 0.8.
Explanation:
1. Use the addition rule of probability: P(A or B) = P(A) + P(B) - P(A and B)
2. P(Chocolate) = 60/100 = 0.6
3. P(Vanilla) = 50/100 = 0.5
4. P(Both) = 30/100 = 0.3
5. P(Chocolate or Vanilla) = 0.6 + 0.5 - 0.3 = 0.8
6. 0.8 = 80%