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Outcome calculation is a vital step in understanding dice probability problems. When dealing with dice, every roll can produce a finite number of results. Each die has 6 faces, numbered 1 through 6. When we roll two dice, we combine these results, leading to multiple possible outcomes. Here, we're interested in the total number of possibilities. For a single roll of two dice, we calculate the total outcomes by multiplying the number of faces on each die. Hence, if both players A and B roll one die each, we get: 6 (faces of die A) × 6 (faces of die B) = 36 possible outcomes.
After establishing the total outcomes, we must count the specific scenarios that fit our criteria—such as B's score being greater than A's score. By systematically working through the combinations, we identify 15 winning outcomes for B, considering all cases where A and B's dice rolls produce varied results. This understanding lays the groundwork for more advanced probability calculations and expected value assessments.

Probability theory is the backbone of analyzing dice rolls. It helps us determine the likelihood of various events occurring, such as one gambler winning over another. The probability of an event is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
For instance, in the dice problem where B wins if his roll exceeds A's roll, the favorable outcomes for B winning are 15 out of 36 total outcomes. Hence, the probability of B winning is \(\frac{15}{36}\) or simplified to \(\frac{5}{12}\).
The odds are another way to express these probabilities. Odds represent the ratio of the number of favorable outcomes to the number of unfavorable outcomes. For A to win, we have calculated, from the step-by-step solution, the odds in A's favor as 7 to 5. This stems from comparing A’s wins and B’s wins: Probability(A’s win) = \(\frac{21}{36}\) (since neutral ties are removed in favor measurements) and Probability(B’s win) = \(\frac{15}{36}\). Adjusting these probabilities to ratio form gives us the winning odds for A as 7 to 5.

Expected value is a key concept that allows players to understand long-term outcomes in gambling scenarios. It represents the average result expected from repeated trials of a random event.
In problem (b), Gambler C's expected outcome when they both roll their dice twice needs analysis, especially with C increasing their stake to £1.10 while D continues with £1.00. To determine the expected value, the steps start with calculating the probabilities of each possible score sum from rolling two dice. These probabilities help in evaluating the chances of C's total score exceeding D's or vice versa.
For each possible total, we derive the probability by considering different combinations that produce the sum. We then use these probabilities to find the likelihood of one total being higher than another. Once the probabilities for all score scenarios are determined, we calculate the expected winnings for each player. The profit for C or D depends on the weighted average, factoring in stake adjustments. If C’s stake adjustment overcompensates for the odds' imbalance, we would conclude C to profit in the long run; otherwise, D would benefit.