Question

we roll a pair of dice two times. find the probability that a total of five is rolled each time. the probability that a total of five is rolled each time is (simplify your answer. type an integer or a fraction.)

Explanation:

Step1: Calculate the total number of outcomes when rolling two dice

When rolling two dice, each die has 6 possible outcomes. So the total number of outcomes when rolling two dice is \(n(S)=6\times6 = 36\) (by the fundamental counting principle).

Step2: Find the number of ways to get a sum of 5

The possible combinations \((x,y)\) such that \(x + y=5\) where \(x\) and \(y\) are the outcomes of the first and second die respectively: \((1,4)\), \((2,3)\), \((3,2)\), \((4,1)\). So \(n(E)=4\).

Step3: Calculate the probability of getting a sum of 5 in a single roll

The probability \(P\) of an event \(E\) is \(P(E)=\frac{n(E)}{n(S)}\). For getting a sum of 5 in a single roll of two dice, \(P_1=\frac{4}{36}=\frac{1}{9}\).

Step4: Use the multiplication rule for independent events

Since the two rolls of the pair of dice are independent events, if \(P_1\) is the probability of getting a sum of 5 in the first roll and \(P_2\) is the probability of getting a sum of 5 in the second roll, and \(P = P_1\times P_2\) (for independent events \(A\) and \(B\), \(P(A\cap B)=P(A)\times P(B)\)). Here \(P_1 = P_2=\frac{1}{9}\), so \(P=\frac{1}{9}\times\frac{1}{9}\)

Answer:

\(\frac{1}{81}\)