12. If you roll three fair six-sided dice, what is the probability that you get two sixes and
one odd number, in any order?
a. 1/36
b. 1/72
c. 1/24
d. 5/9
Probability Two Sixes One Odd Three Dice: Exact Calculation & Explanation
Three fair six-sided dice yield 216 total outcomes. Favorable cases occur with exactly two sixes and one odd number (1, 3, or 5), which can happen in 3 positions for the odd die, giving 3 × 3 = 9 outcomes. Thus, the probability simplifies to 9/216 = 1/24.
Detailed Solution
Each die has faces {1,2,3,4,5,6}, with odds {1,3,5} (probability 1/2) and sixes fixed at one face. The scenario requires exactly two sixes (one specific face) and one odd (three choices), in any order. Choose the odd position (3 ways: first, second, or third die), then assign: 3 odds × 1 × 1 for sixes per case, totaling 9 favorable. Probability = 9/216 = 1/24 after dividing numerator and denominator by 9.
Option Analysis
- a. 1/36: Incorrect; matches single six probability (1/6 × 5/6 × 5/6 adjusted wrongly), undercounts odds and positions.
- b. 1/72: Incorrect; halves the correct value, possibly forgetting 3 odd choices or miscounting positions.
- c. 1/24: Correct; matches 9/216 simplification via greatest common divisor 9.
- d. 5/9: Incorrect; vastly overestimates, perhaps confusing with all-odd or unrelated events.
The probability of two sixes and one odd number with three fair six-sided dice is a classic dice probability question often seen in exams like CSIR NET quantitative aptitude sections. This scenario tests combinations and basic probability, yielding exactly 1/24 as the answer among options. Understanding it builds skills in counting favorable outcomes over total possibilities (216 for three dice).
Step-by-Step Calculation
Total outcomes: 63 = 216. Favorable: Exactly two 6s (one face each) and one odd (1,3,5: three choices). Positions for odd: 3 choices. Per position: 3 × 1 × 1 = 3 ways. Total favorable: 9. Probability: 9/216 = 1/24.
Key Figures
| Aspect | Details |
|---|---|
| Total Outcomes | 216 |
| Favorable Outcomes | 9 (3 positions × 3 odds) |
| Correct Option | c. 1/24 |
| Common Error | Forgetting 3 odd choices (yields 1/72) |
Why Other Options Fail
Option a (1/36) ignores position variety; option b (1/72) misses odds count; option d (5/9) confuses unrelated events. Master this for dice problems by always listing cases or using binomial coefficients: C(3,2) × (1/6)2 × (3/6) = 3 × 1/36 × 1/2 = 1/24.
