Four die are thrown simultaneously. The probability that 4 and 3 appear on two of the die given that 5 and 6 have appeared on other two die is:
Step-by-Step Solution
Key Concept: This is a conditional probability problem requiring inclusion-exclusion principle to count outcomes where 5 and 6 appear on exactly two dice, then count favorable outcomes where 4 and 3 appear on the remaining two dice. The denominator uses |A ∪ B| = 6^4 - 2·5^4 + 4^4 to exclude cases where 5 or 6 don't appear.
When 5 and 6 appear on two dice, the sample space reduces to $6^4 - 2 imes 5^4 + 4^4$ by inclusion-exclusion. With 24 favorable outcomes, the probability is $ rac{24}{302} = rac{12}{151}$.
Correct Answer: 3