Question

$3.8$ (rolling dice, part 1). Two regular (6-sided, fair) dice are rolled. Write down the entire sample space $5$, like we did in class; what is $\#(5)$, the number of elements in $S$ ? Now, let $A$ be the event that the sum of the faces is odd, and $B$ be the event that at least one six is rolled. Describe in words the events $A \cap B$, $A \cup B$, and $A \cap B^{C}$. Assuming that all the outcomes are equally likely, compute $PCA), P(B), PA \cap $, $P(A cup B), P\left( \cap B^{c}\right) $. SP.SD.606

Answer 1

The sample space (S) for rolling two regular 6-sided dice is given by (S = {(1,1), (1,2), ..., (6,6)}, and (#(S) = 36). For the events
`\(A \cap B\)`,
`\(A \cup B\)`, and
`\(A \cap B^(C)\)`, they represent the outcomes where the sum of the faces is odd and at least one six is rolled, the sum of the faces is odd or at least one six is rolled, and the sum of the faces is odd while no six is rolled, respectively. The probabilities are
`\(P(A \cap B) = (11)/(36)\), \(P(A \cup B) = (26)/(36)\)`, and
`\(P(A \cap B^(C)) = (5)/(12)\). Also, \(P(A) = (18)/(36)\), \(P(B) = (11)/(36)\), and \(P(B^(C)) = (25)/(36)\).`

Step-by-step explanation:

The sample space (S) for rolling two regular 6-sided dice consists of all possible outcomes, and #(S) denotes the number of elements in this set. In this case, each die has 6 faces, so there are (6 ×6 = 36) possible outcomes, ranging from (1,1) to (6,6).

Events
`\(A \cap B\)`,
`\(A \cup B\)`, and
`\(A \cap B^(C)\)` involve the intersection, union, and complement of events (A) and (B).
`\(A \cap B\)` represents the outcomes where both events (A) and (B) occur,
`\(A \cup B\)` represents the outcomes where either event (A) or (B) or both occur, and
`\(A \cap B^(C)\)` represents the outcomes where event (A) occurs while event (B) does not.

The probabilities are calculated based on the number of favorable outcomes divided by the total number of outcomes. For example,
`\(P(A \cap B) = (\#(A \cap B))/(\#(S)) = (11)/(36)\)`, where
`\(\#(A \cap B)\)` is the number of outcomes in the intersection of events (A) and (B). Similarly,
`\(P(A \cup B)\)` is the probability of the union of events (A) and (B), and
`\(P(A \cap B^(C))\)` is the probability of the intersection of event (A) and the complement of event (B).

In summary, these probabilities provide insights into the likelihood of different events occurring when rolling two dice, considering both odd sums and the presence of at least one six.