Two fair dice are rolled. Let X equal the product of the 2 dice, compute
� = �
for � = 1,2,…,36
5/36
To compute the probability distribution of the random variable X, which represents the product of two fair dice, we need to determine the probability of each possible outcome. Since each die has 6 sides numbered 1 to 6, there are a total of 36 possible outcomes when two dice are rolled.
Let's break down the problem into steps:
Step 1: Determine the sample space
The sample space is the set of all possible outcomes. In this case, the sample space is the set of all pairs of numbers that can result from rolling two dice. This can be represented as {(1,1), (1,2), ..., (6,6)}.
Step 2: Determine the total number of outcomes
The total number of outcomes is equal to the size of the sample space. In this case, there are 36 outcomes since there are 6 options for the first die and 6 options for the second die, giving a total of 6*6=36 outcomes.
Step 3: Calculate the probability for each outcome
To calculate the probability of each outcome, we need to determine how many favorable outcomes there are for each value of X (the product of the dice).
For X = 1, the only possible outcome is (1,1), so there is 1 favorable outcome.
For X = 2, there are two possible outcomes: (1,2) and (2,1). So, there are 2 favorable outcomes.
Similarly, we can determine the number of favorable outcomes for each value of X:
X = 1: 1 favorable outcome
X = 2: 2 favorable outcomes
X = 3: 2 favorable outcomes
X = 4: 3 favorable outcomes
X = 5: 4 favorable outcomes
X = 6: 5 favorable outcomes
X = 7: 6 favorable outcomes
X = 8: 5 favorable outcomes
X = 9: 4 favorable outcomes
X = 10: 3 favorable outcomes
X = 11: 2 favorable outcomes
X = 12: 1 favorable outcome
Step 4: Calculate the probabilities
To calculate the probability for each value of X, divide the number of favorable outcomes for that value by the total number of outcomes.
For example, for X = 4:
P(X = 4) = Number of favorable outcomes for X = 4 / Total number of outcomes = 3/36 = 1/12
Following the same calculation for each value of X, we can find the probabilities for all the values of X ranging from 1 to 12.
Thus, the probability distribution � = � for � = 1,2,…,36 would be:
�(1) = 1/36
�(2) = 2/36 = 1/18
�(3) = 2/36 = 1/18
�(4) = 3/36 = 1/12
�(5) = 4/36 = 1/9
�(6) = 5/36
�(7) = 6/36 = 1/6
�(8) = 5/36
�(9) = 4/36 = 1/9
�(10) = 3/36 = 1/12
�(11) = 2/36 = 1/18
�(12) = 1/36