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The problem asks us to find the probability of rolling at least one 4 when rolling two dice, and then extend this to n dice. To solve this, we need to understand the concept of complementary probability and independent events.
Complementary Probability: The probability of an event happening is 1 minus the probability of it not happening.
P(A)=1−P(A′)
Independent Events: The outcome of one dice roll does not affect the outcome of another dice roll.
Single Dice:
Two Dice:
Probability of not rolling a 4 on both dice: (65)2=3625
Probability of rolling at least one 4:
P(at least one 4)=1−P(no 4 on both dice)=1−3625=3611
Probability of not rolling a 4 on a single die: 65
Probability of not rolling a 4 on all n dice: (65)n
Probability of rolling at least one 4:
P(at least one 4)=1−(65)n
Complementary Probability: By calculating the probability of the complementary event (not rolling a 4), we can easily find the probability of the event of interest (rolling at least one 4).
Independence: Each dice roll is independent, allowing us to multiply probabilities across dice.
Generalization: Extending from two dice to n dice involves using the same logic but applying it across more dice, resulting in the formula 1−(65)n.
This approach efficiently calculates the probability of rolling at least one 4, leveraging the simplicity of complementary probability and the independence of dice rolls.