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Data Interview Question

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Solution & Explanation

Understanding the Problem

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.

Key Concepts

  1. Complementary Probability: The probability of an event happening is 1 minus the probability of it not happening.

    P(A)=1P(A)P(A) = 1 - P(A')

  2. Independent Events: The outcome of one dice roll does not affect the outcome of another dice roll.

Solution Steps

1. Probability with Two Dice
  • Single Dice:

    • Probability of rolling a 4: P(4)=16P(4) = \frac{1}{6}
    • Probability of not rolling a 4: P(not 4)=116=56P(\text{not } 4) = 1 - \frac{1}{6} = \frac{5}{6}
  • Two Dice:

    • Probability of not rolling a 4 on both dice: (56)2=2536\left(\frac{5}{6}\right)^2 = \frac{25}{36}

    • Probability of rolling at least one 4:

      P(at least one 4)=1P(no 4 on both dice)=12536=1136P(\text{at least one 4}) = 1 - P(\text{no 4 on both dice}) = 1 - \frac{25}{36} = \frac{11}{36}

2. Extending to n Dice
  • n Dice:
    • Probability of not rolling a 4 on a single die: 56\frac{5}{6}

    • Probability of not rolling a 4 on all n dice: (56)n\left(\frac{5}{6}\right)^n

    • Probability of rolling at least one 4:

      P(at least one 4)=1(56)nP(\text{at least one 4}) = 1 - \left(\frac{5}{6}\right)^n

Explanation

  • 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(56)n1 - \left(\frac{5}{6}\right)^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.