Week 2 Assignment 1: What Is The Probability Of Rolling A Fo

Week 2 Assignment1 A What Is The Probability Of Rolling A Four In Th

What is the probability of rolling a four in the gambling dice game of craps (given two six-sided dice)? What is the probability that a player can roll a four three times in a row (assuming each roll is independent)?

Paper For Above instruction

The probability of rolling a four with two six-sided dice in the game of craps can be determined by examining the total number of possible outcomes. Since each die has six faces, the total number of outcomes when rolling two dice is 6 × 6 = 36. To obtain a sum of four, the possible combinations are (1,3), (3,1), and (2,2). There are 3 such outcomes. Therefore, the probability of rolling a four in a single roll is 3/36, which simplifies to 1/12.

To find the probability of rolling a four three times in a row, assuming each roll is independent (the outcome of one roll does not influence the others), we raise the single-event probability to the third power. Since the probability of rolling a four on one roll is 1/12, the probability of doing so three consecutive times is (1/12)³ = 1/1728. This means that in a sequence of independent rolls, the likelihood of achieving three consecutive fours is relatively low, emphasizing the rare occurrence of such an event in the game.

Understanding these probabilities is fundamental in the analysis of gambling games, where players, researchers, and statisticians evaluate the fairness and expected outcomes of dice rolls. The calculation underscores the importance of probability theory in predicting and assessing risks associated with chance-based activities. Additionally, these probabilities are rooted in basic combinatorial principles, illustrating how simple mathematical concepts underpin real-world gaming scenarios.

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