Roll a pair of dice - find probability at least one die shows a given number.

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Question 1

Two fair six-sided dice are rolled.

What is the probability of at least one die showing a $\var{number}$ ?

Probability = interpreted as $\displaystyle{}$ Expected answer:interpreted as $\displaystyle{\frac{ 11 }{ 36 }}$

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Let $A$ be the event that first dice shows a $\var{number}$ $\Rightarrow P(A)=\frac{1}{6}$ .

Let $B$ be the event that second dice shows a $\var{number}$ $\Rightarrow P(B)=\frac{1}{6}$ .

$A$ and $B$ are independent events so $P(A\cap B) = P(A)\times P(B)$ .

We want the probability $P(A \cup B)$ of either $A$ or $B$ showing $\var{number}$ and

$\begin{eqnarray*} P(A \cup B) &=& P(A)+P(B)-P(A \cap B)\\ &=& P(A)+P(B)-P(A)P(B)\\ &=&\frac{1}{6}+ \frac{1}{6}-\frac{1}{36}\\ &=& \frac{11}{36} \end{eqnarray*}$

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Roll a pair of dice - find probability at least one die shows a given number.

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