The question: A pair of fair dice is thrown 10 times. What is the probability of obtaining a double six in at least two throws?
My attempt: Let X denote the total number of double sixes obtained. On any throw, the probability of obtaining a double six is $\frac 1{36}$ and thus, the probability of not obtaining a double six is $\frac {35}{36}$.
$P(X=1) = 10(\frac 1{36})(\frac {35}{36})^9$ since the double six can be obtained in any of the 10 throws.
$P(X=0) = (\frac {35}{36})^{10}$
So $P(X \ge 2) = 1 – P(X=1) – P(X=0) = 1 – (\frac {10}{36})(\frac{35}{36})^9 – (\frac{35}{36})^{10}$
However, the school's solution says that $P(X \ge 2) = 1 – (\frac {1}{36})(\frac{35}{36})^9 – (\frac{35}{36})^{10}$
Which is correct? And why?
Best Answer
Your answer is correct. The school’s solution is in error.
$\begin{align} \mathsf P(X\geq 2) & = 1 - \mathsf P(X\leq 1) & \text{By the Law of Complements} \\ & = 1 - {10\choose 0}p^0(1-p)^{10} - {10\choose 1}p^1(1-p)^9 & \text{because it is a binomial distribution} \\ & = 1- \frac{35^{10}}{36^{10}}-\frac{10\cdot 35^9}{36^{10}} \end{align}$