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- Apr 14, 2013

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A wheel of fortune is divided into $10$ equal sized parts, of which one of them brings the jackpot.

- A player wants to investigate the regularity of the wheel of fortune. He turns the wheel 20 times. Calculate the probability that he will get at least one main prize if it is a laplace wheel.
- How often does the player have to spin the wheel of fortune to get at least one main prize with a probability of at least 90%.
- The player objects to the wheel when less than twice the jackpot appears at 20 times turn. Calculate the probability that the wheel will be misjudged.
- What does the player have to change in his test so that the significant level is about $6\%$ ?

I have done the following:

- $$P(X\geq 1)=1-P(X\leq 1)=1-\sum_{i=0}^1\binom{10}{i}\left (\frac{1}{10}\right )^{20}\cdot \left (1-\frac{1}{10}\right )^{20-i}$$

$$ $$

- \begin{align*}P(X\geq k)\geq 90\% &\Rightarrow 1-P(X\leq k)=90\% \\ & \Rightarrow 1-\sum_{i=0}^k\binom{10}{i}\left (\frac{1}{10}\right )^{20}\cdot \left (1-\frac{1}{10}\right )^{20-i}\geq 0.9 \\ & \Rightarrow \sum_{i=0}^k\binom{10}{i}\left (\frac{1}{10}\right )^{20}\cdot \left (1-\frac{1}{10}\right )^{20-i}\leq 0.1\end{align*}

Is everything correct so far?

Could you give me a hint for the other two questions?