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In probability, the sample space is the set of all possible outcomes that can occur in an experiment. When tossing two dice, each die has six faces, numbered from 1 to 6. This gives us a simple way to represent the sample space using ordered pairs, where the first number in the pair corresponds to the outcome of the first die, and the second number corresponds to the outcome of the second die. For example, the pair \((1, 2)\) denotes the first die showing a 1 and the second die showing a 2.

With two dice, there are 6 possible outcomes from the first die and 6 from the second, combining to create a total of \(6 \times 6 = 36\) possible pairs or outcomes in the sample space. Here’s a small glimpse of the possibilities you’ll see:

• \((1, 1)\)
• \((2, 3)\)
• \((4, 5)\)
• \((6, 6)\)

Understanding this structured list of possibilities sets the stage for determining probabilities of certain events happening.

Calculating the probability of an event in our dice experiment involves counting the number of favorable outcomes and dividing by the total number of outcomes in the sample space.

### Probability of a Specific SumConsider finding the probability where the sum of numbers shown on both dice equals 7. We count the pairs that add up to 7:

• \((1, 6)\)
• \((2, 5)\)
• \((3, 4)\)
• \((4, 3)\)
• \((5, 2)\)
• \((6, 1)\)

There are 6 such outcomes, so the probability is \(\frac{6}{36} = \frac{1}{6}\). To find the probability of a sum of 11, look for outcomes like \((5, 6)\) and \((6, 5)\), giving us a probability of \(\frac{2}{36} = \frac{1}{18}\).

### Probability of Rolling DoublesDoubles occur when both dice show the same number. The possible outcomes are \((1, 1)\), \((2, 2)\), \(\ldots\), and \((6, 6)\). With 6 outcomes, the probability of rolling doubles is \(\frac{6}{36} = \frac{1}{6}\).

These probability calculations provide insights into the likelihood of certain events when rolling two dice.

Each die can land on one of 6 numbers, and the total outcomes when considering two dice improve our understanding of possible results. Key outcomes come into play in this probability exercise:

### Odds and DoublesWhen both dice show an odd number like 1, 3, or 5, there are 9 possible outcomes. Here they are listed:

• \((1, 1)\)
• \((1, 3)\)
• \((1, 5)\)
• \((3, 1)\)
• \((3, 3)\)
• \((3, 5)\)
• \((5, 1)\)
• \((5, 3)\)
• \((5, 5)\)

This results in a probability calculation of \(\frac{9}{36} = \frac{1}{4}\) for both dice showing odd numbers.

By understanding the outcomes and probabilities associated with each scenario, you can better grasp how structured randomness contributes to probability theory in dice rolls. Focusing on outcomes like odd numbers or doubles helps illuminate specific target outcomes and their probabilities.