Final answer:
The probability that the player does not get to begin the game of Parcheesi on either the first or the second roll is 625/1296, calculated by multiplying the probability of not starting on a single roll by itself, as the rolls are independent events.
Step-by-step explanation:
The student is asking about the probability of not being able to start a game of Parcheesi on the first or second roll of a pair of dice. In order to find this probability, we first need to determine the probability of starting the game, which occurs if at least one die shows a five, or if the sum of both dice is five.
Probability of Starting the Game
To begin, the probability of rolling a five on a single die is 1/6, since there are six possible outcomes and only one of them is a five. For two dice, there are 36 possible combinations (6 sides on the first die × 6 sides on the second die). The favorable outcomes for starting the game are when we have a five on the first die (6 possibilities), a five on the second die (another 6 possibilities), and when the sum is five without either die being a five (which is only one case, 2 and 3). However, since rolling a five on both dice (5,5) is counted twice in our initial count, we need to subtract one to avoid double counting. Therefore, there are 11 favorable outcomes.
Probability of starting the game (P(start)) = Number of favorable outcomes / Total possible outcomes = 11/36
Probability of not starting the game (P(not start)) = 1 - P(start) = 25/36
Probability of Not Starting on the First or Second Roll
Since the rolls are independent events, the probability of not starting the game on both the first and second roll is the product of the individual probabilities:
Probability of not starting on either the first or second roll = P(not start) × P(not start) = (25/36) × (25/36) = 625/1296
Therefore, the probability that the player does not get to begin the game on either the first or the second roll is 625/1296.