Question

What is the most likely outcome when we throw two fair dice, i.e., what is the most likely sum that the two dice would add to? Why? This problem can be solved by first principles. The probability P(E) for an event E is the ratio |E|/|S|, where |E| is the cardinality of the event space and |S| is the cardinality of the sample space. For example, when we throw a fair die, the event space is S = {1,2,3,4,5,6} and its cardinality is |S| = 6. So if we ask, what is the likelihood that the roll of a fair dice will be a prime number, we are essentially asking what is the likelihood to roll one of these numbers: {1,2,3,5}. This set is the event space E and its cardinality |E| = 4. Now we can compute the probability:
P(die rolls a prime number) =
|{1,2,3,5}| |{1,2,3,4,5,6}|
=
4 6
=
2 3
= 66%
When we throw two fair dice, their sum is a number between 2 and 12. These sums can be unique, e.g., the sum of 2 can be obtained only when both dice roll to 1. But the sum of 4 can be obtained with three difference combinations: 1 + 3, 2 + 2, and 3+1. In other words, there is only one event that leads to a sum of 2, but three events that lead to a sum of 4.

1. (3 points) What is the most likely outcome when we throw two fair dice , i.e., what is the most likely sum that the two dice would add to? Why? This problem can be solved by first principles. The probability P(E) for an event E is the ratio lEI/SI, where JEl is the cardinality of the event space and iSI is the cardinality of the sample space. For example, when we throw a fair die, the event space is S = {1, 2, 3, 4, 5,6) and its cardinality is IS = 6" So if we ask, what is the likelihood that the roll of a fair dice will be a prime number, we are essentially asking what is the likelihood to roll one of these numbers: (1,2,3,5]. This set is the event space E and its cardinality IEI-4. Now we can compute the probability: |{ 1, 2, 3, 51-4=-=66% 旧,2, 3, 4, 5, 61 6 3 P(die rolls a prime number) When we throw two fair dice, their sum is a number between 2 and 12. These sums can be unique, e.g., the sum of 2 can be obtained only when both dice roll to 1. But the sum of 4 can be obtained with three difference combinations: 1 3, 22, and 3+1. In other words, there is only one event that leads to a sum of 2, but three events that lead to a sum of 4.

Answer 1

(い) (か) ( Χ.Α) ) 5 5

1 0 (5,5) →초 t久 (61 )T rary 36 36 6 H6 my meny