Points to Remember:
- Probability is the likelihood of an event occurring.
- The sample space for throwing two dice is 36 possible outcomes.
- We need to identify the favorable outcomes where both dice show the same digit.
Introduction:
This question requires a factual and analytical approach to determine the probability of a specific event in a dice-throwing experiment. Probability is a mathematical measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. When two dice are thrown, the possible outcomes are numerous, and calculating the probability of a specific outcome requires understanding the sample space and the favorable outcomes.
Body:
1. Defining the Sample Space:
When two dice are thrown simultaneously, each die can show any number from 1 to 6. The total number of possible outcomes (the sample space) is 6 x 6 = 36. This can be represented as ordered pairs (x, y), where x is the outcome of the first die and y is the outcome of the second die. For example, (1,1), (1,2), (1,3), …, (6,6) are all possible outcomes.
2. Identifying Favorable Outcomes:
The question asks for the probability of getting the same digit on both dice. This means we are looking for outcomes where x = y. The favorable outcomes are: (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). There are 6 such outcomes.
3. Calculating the Probability:
Probability is calculated as the ratio of favorable outcomes to the total number of possible outcomes. In this case:
Probability (same digit) = (Number of favorable outcomes) / (Total number of possible outcomes) = 6/36 = 1/6
Therefore, the probability of getting the same digit on the upper faces of both dice is 1/6 or approximately 0.167.
Conclusion:
In conclusion, the probability of obtaining the same digit on both dice when two dice are thrown simultaneously is 1/6. This calculation involved identifying the sample space (36 possible outcomes) and the favorable outcomes (6 outcomes where both dice show the same number). The probability was then calculated as the ratio of favorable outcomes to the total number of possible outcomes. This simple example demonstrates the fundamental principles of probability theory, which has wide-ranging applications in various fields, from risk assessment to statistical analysis. Understanding probability helps us make informed decisions based on the likelihood of different events occurring. Further exploration into probability distributions and statistical inference can provide a more comprehensive understanding of uncertainty and decision-making under conditions of risk.