Two "Loaded" Dice each ahve the property that a 2 or a 1 is three times as likely to appear as a 1, 3, 5, or 6 on each roll. What is the probabilitiy that a 7 will be the sum when the two dice are rolled?
To find the probability that a sum of 7 will be obtained when two dice are rolled, we need to consider all possible ways in which the dice can add up to 7. Let's break it down step by step.
Step 1: Determine the probability of each outcome for each individual die.
Since each die is loaded, the probabilities of rolling a particular number are not equal. If a 2 or a 1 is three times as likely to appear as a 1, 3, 5, or 6, we can assign probabilities as follows:
P(1 or 2) = 3x * P(3, 5, or 6)
P(3, 5, or 6) = P(1 or 2) / 3x
Note: The value of 3x can vary, but let's assume it is a constant value for both dice.
Step 2: Determine the probability of obtaining the sum of 7.
To find the sum of 7, we need the following combinations:
(1, 6) or (6, 1)
(2, 5) or (5, 2)
(3, 4) or (4, 3)
Step 3: Calculate the probability of each combination.
Let's assume the probability of rolling any number on the loaded dice is denoted by P(n), where n represents the number rolled.
For combinations (1, 6) and (6, 1):
P(1, 6) = P(1) * P(6) = P(1) * P(1/3x)
P(6, 1) = P(6) * P(1) = P(1/3x) * P(1)
For combinations (2, 5) and (5, 2):
P(2, 5) = P(2) * P(5) = P(1/3x) * P(1/3x)
P(5, 2) = P(5) * P(2) = P(1/3x) * P(1/3x)
For combinations (3, 4) and (4, 3):
P(3, 4) = P(3) * P(4) = P(1/3x) * P(1/3x)
P(4, 3) = P(4) * P(3) = P(1/3x) * P(1/3x)
Step 4: Calculate the total probability of obtaining a sum of 7.
To find the probability of each combination, we need to multiply the individual probabilities together and sum them up for all valid combinations:
P(7) = P(1, 6) + P(6, 1) + P(2, 5) + P(5, 2) + P(3, 4) + P(4, 3).
Now, you'll need to know the specific value of 3x (given in the problem statement) in order to calculate the probabilities. Once you have that value, substitute it into the equations above and calculate the final probability.