Two balanced dice are rolled. Let x be the product of the two dice.
3.1. Construct the probability distributions of x.
1,1 = 1
1,2 = 2
1,3 = 3
1,4 = 4
1,5 = 5
1,6 = 6
do the same for 2 all the way to 6.
To do the prob distrib. take the total number of products (should be 36)
1, 1 = 1 happens only once 1/36
To construct the probability distribution of the product x of two balanced dice, we need to determine the probability of each possible outcome.
Step 1: Determine the Sample Space:
The sample space consists of all possible outcomes when two dice are rolled. Since each die has six sides numbered 1 to 6, there are a total of 6 * 6 = 36 possible outcomes.
Step 2: Determine the Possible Products:
The possible products x can range from 1 to 36, as the minimum product is 1 (1 * 1) and the maximum product is 36 (6 * 6).
Step 3: Calculate the Probability for each Product:
To calculate the probability for each product, we need to determine the number of ways each product can be obtained and divide it by the total number of possible outcomes.
Here's how you can calculate the probability for each product:
Product x = 1: There is only one combination that results in a product of 1, which is (1,1). So the probability P(x=1) = 1/36.
Product x = 2: To obtain a product of 2, you can either roll a 1 and a 2 or a 2 and a 1. So the probability P(x=2) = 2/36 = 1/18.
Similarly, by considering all possible combinations and calculating their probabilities, you can find the probabilities for the remaining products.
Product x = 3: P(x=3) = 2/36 = 1/18.
Product x = 4: P(x=4) = 3/36 = 1/12.
Product x = 5: P(x=5) = 4/36 = 1/9.
Product x = 6: P(x=6) = 5/36.
Product x = 7: P(x=7) = 6/36 = 1/6.
Product x = 8: P(x=8) = 5/36.
Product x = 9: P(x=9) = 4/36 = 1/9.
Product x = 10: P(x=10) = 3/36 = 1/12.
Product x = 11: P(x=11) = 2/36 = 1/18.
Product x = 12: P(x=12) = 1/36.
This completes the probability distribution for the product x of two balanced dice.