Question

Two six-sided dice are thrown. The outcome whenever a single die is thrown is 1,2,3,4,5, or 6 , and all these numbers are equally-likely. Answer each part below by typing your answer directly into the text box. Express fractions as a/b. You may use the equation editor if you wish, but this is not required. Give a brief explanation of how you made your calculations. Part I (4 points). Compute the probability that the sum of the two dice is exactly 4. Part II (4 points). Compute the probability that neither die shows a 3 . Hint: Complement rule! Part III (4 points). Compute the conditional probability that the sum of the two dice is 12 , given that one of the dice shows a 6 .

Answer 1

The probability that the sum of two dice is exactly 4 is 1/12. The probability that neither die shows a 3 is 25/36. The conditional probability that the sum of two dice is 12, given that one die shows a 6, is 1/6.

Step-by-step explanation:

To calculate the probability that the sum of the two dice is exactly 4 (Part I), we consider the possible pairs that give this sum: (1,3), (2,2), and (3,1). There are 3 favorable outcomes out of 36 possible outcomes when two dice are rolled (6 for each die), so the probability is 3/36 or 1/12.

For Part II, the probability that neither die shows a 3 is the complement of at least one die showing a 3. Each die has a 5/6 chance of not rolling a 3, so we multiply these probabilities to get 25/36.

In Part III, the conditional probability that the sum is 12 given one die shows a 6 involves recognizing that the only way to get a sum of 12 is if the other die also shows a 6. There is 1 favorable outcome out of 6 possible outcomes of the other die, making the probability 1/6.