$N\in \mathbb{N}$ christmas presents will be distributed. In every gift there is additional either a chocolate snowman or a chocolate reindeer. It will be independent of each other and with the same probability of giving out gifts with a chocolate snowman or chocolate reindeer. So let $X_1, \ldots , X_n$ be independent and Bernoulli-distributed with probability of success $p =\frac{1}{2}$. The event $\{X_j = 1\}$ means that in the $j$th present there is a chocolate reindeer. Let $a_n$ be the probability that at least 60% of the $n$ gifts distributed contains a chocolate reindeer.
(a) Calculate $a_{10}$ explicitly. Enter intermediate steps.
(b) Use the inequality $\displaystyle{P\left [\left |\frac{1}{n}\sum_{j=1}^nX_j-p\right |\geq \epsilon\right ]\leq \frac{1}{4n\epsilon^2}}$ from Bernoulli's weak law of large numbers, to estimate $a_{100}$. Does it hold $a_{100}<a_{10}$ ?
(c) Determine $\displaystyle{\lim_{n\rightarrow \infty}a_n}$.
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What exactly does $a_n$ mean? Is it the probability $P\left [X_1, \ldots , X_n\geq 0.6\cdot n\right ]$ ?