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Rolling a fair die means each face (or side) of the die has an equal chance of landing face up. A standard six-sided fair die has numbers from 1 to 6, and each number is just as likely to appear as any other. This is because a fair die is unbiased, meaning there are no adjustments or weights that could favor one outcome.

When you roll a fair die, the probability of any specific number appearing, such as a 6, is calculated as the ratio of favorable outcomes (the single side showing a 6) to the total number of outcomes (all 6 sides). Thus, the probability of rolling a 6 is \( \frac{1}{6} \). Similarly, the probability of not rolling a 6 (getting either 1, 2, 3, 4, or 5) is \( \frac{5}{6} \).

• Every side has a cooperative chance.
• Total possible outcomes when rolling once: 6.
• Probability of each number being rolled: \( \frac{1}{6} \).

In probability theory, an event is said to be independent if the outcome of one event does not affect the outcome of another. This is a key concept, especially when dealing with games of chance like dice rolls, where each roll is independent.

Consider rolling a fair die multiple times. The result of one roll does not influence the result of any subsequent rolls. This means that the probability of rolling a 6 on your next roll remains \( \frac{1}{6} \), whether or not you rolled a 6 previously.

• No past roll affects future rolls.
• The probability remains constant.

This independence allows us to easily calculate the probability of a sequence of events by multiplying their individual probabilities. For example, if we wanted the probability of not rolling a 6 three times in a row, we'd calculate \( \left( \frac{5}{6} \right)^3 \) because each roll has a consistent probability of \( \frac{5}{6} \).

This consistency is why independent event calculations are foundational in probability theory.

Sequential probability involves determining the likelihood of a series of events happening one after the other. In this context, it often means calculating the probability of specific outcomes occurring in a particular order.

Consider the exercise where each person takes turns rolling a die, trying to be the first to roll a 6. When we calculate the probability of Person B throwing the first 6 on her second throw (the fourth roll overall), we look at the sequence: three rolls that aren't 6, followed by a roll that is 6.
The key steps to solving such problems involve:

• Identifying the required sequence of outcomes for the specific scenario.
• Calculating the probability of each event occurring in the desired order.
• Combining these probabilities to find the overall sequence probability.

In Person B's scenario, the sequence we need is "not 6" (three times), followed by "6" (on the fourth roll). The computation involves exponentiation for the repeating outcomes and a straightforward multiplication with the final probability.
So, the calculation for the first three rolls not being a 6 is \( \left( \frac{5}{6} \right)^3 \), and the probability of the fourth roll being a 6 is \( \frac{1}{6} \). Therefore, multiplying these gives us the probability of the sequence we wanted: \( \frac{125}{1296} \).

This method systematically determines the chance of a specific order of events unfolding, reflecting the nature of sequential probability.