Two Dice Are Rolled Independently And The Two Numbers That T

14 5 Two Dice Are Rolled Independently And The Two Numbers That Turn U

Determine the probabilities related to two independent dice rolls, where the two numbers that turn up are recorded. Specifically: (a) the probability that both numbers are odd and less than 5; (b) the probability that the sum of the two numbers equals 10; and (c) the probability that the two numbers differ by at least 3.

Paper For Above instruction

Rolling two dice independently forms a classic problem in probability theory, allowing for a comprehensive analysis of joint outcomes based on the properties of individual dice faces. Each die has six faces numbered from 1 to 6, and the independence assumption means each pair of outcomes occurs with equal probability of 1/36. This paper examines three specific probability queries based on the outcomes of such an experiment.

The first part investigates the probability that both numbers are odd and less than 5. On a single die, the odd numbers less than 5 are 1 and 3. When two dice are rolled independently, the favorable outcomes for each die are therefore (1,1), (1,3), (3,1), and (3,3). These outcomes collectively form four favorable pairs. Since each pair has a probability of 1/36, the combined probability that both dice display such outcomes is 4/36, which simplifies to 1/9. This result hinges on the independence and uniformity assumptions, emphasizing that the likelihood of both dice landing on odd numbers less than 5 is relatively low but precisely quantifiable in this systematic manner.

The second question pertains to the probability that the sum of the two dice is exactly 10. The pairs that sum to 10 are (4,6), (5,5), and (6,4). Each of these outcomes also has a probability of 1/36. Since there are three such outcomes, the probability that the sum is 10 is 3/36, simplifying to 1/12. This calculation underscores the combinatorial approach, where each favorable outcome's probability is summed to find the total probability of a particular sum.

The third part estimates the probability that the two numbers differ by at least 3. The possible pairs with a difference of at least 3 include (1,4), (1,5), (1,6), (2,5), (2,6), (3,6), and their symmetric counterparts with roles reversed. Counting these, there are 12 outcome pairs: (1,4), (1,5), (1,6), (2,5), (2,6), (3,6), and their reversals like (4,1), (5,1), (6,1), etc. Each has a probability of 1/36, summing to 12/36, reducing to 1/3. This analysis demonstrates how the probability of a substantial difference between dice outcomes can be systematically derived through enumeration and symmetry, reinforcing fundamental combinatorial principles in probability.

In summary, the problem explores fundamental aspects of probability concerning independent dice throws, illustrating how combinatorial enumeration and straightforward calculations yield precise outcomes for outcomes with specific properties. These basic methods underpin much more complex probabilistic modeling used in various fields, including game theory, risk analysis, and decision-making processes.

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