Question

a pair of standard dice are rolled. find the probability of rolling a sum of 10 with these dice. p(d₁ + d₂ = 10) = ? / ? be sure to reduce.

Explanation:

Step1: Calculate total number of outcomes

When rolling two dice, each die has 6 possible outcomes. So the total number of outcomes is \(6\times6 = 36\) (by the fundamental counting principle \(n = n_1\times n_2\), where \(n_1 = 6\) (outcomes of first die) and \(n_2=6\) (outcomes of second die)).

Step2: Find the number of favorable outcomes

Let the outcomes of the first die be \(x\) and the second die be \(y\). We want \(x + y=10\) where \(1\leq x\leq6\) and \(1\leq y\leq6\).
The pairs \((x,y)\) that satisfy \(x + y = 10\) are \((4,6)\), \((5,5)\), \((6,4)\). So the number of favorable outcomes \(m = 3\).

Step3: Calculate the probability

The probability formula is \(P=\frac{m}{n}\). Substituting \(m = 3\) and \(n = 36\), we get \(P=\frac{3}{36}=\frac{1}{12}\) (by simplifying the fraction \(\frac{3\div3}{36\div3}\)).

Answer:

\(\frac{1}{12}\)