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On Alan’s first visit to Macau, he decided to try his luck at the Venetian Casino Resort. He played a game that is supposed to offer a 50% chance of winning. Out of seven independent plays, he won the first 6 games and lost the last game. His friend Brad who always bets against Alan is unhappy and thinks the game has been rigged.

State Brad's null and alternative hypothesis

If the null hypothesis is true, what is the chance of winning 6 games in a row and losing the next one?

Calculate the P value based on Alan’s data, to one significant figure.

What can Brad conclude?

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closed as off-topic by Guy Fsone, Maria Mazur, user357151, Misha Lavrov, Rolf Hoyer Nov 25 '17 at 2:34

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  • $\begingroup$ Welcome to MSE! How many chances are there to win 6 games in a row out of 7? $\endgroup$ – QFi Nov 24 '17 at 20:05
  • $\begingroup$ Please read How to ask a homework question. Don't just copy and paste your question here; especially refrain from demanding that we "prove", "show", or "calculate" something. Show your work and say where you got stuck. I'm voting to close this question until you do. $\endgroup$ – Misha Lavrov Nov 25 '17 at 1:43
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Consider the hypothesis:

$H_0$: the coin is unbiased

$H_1$ the coin is biased ($p>1/2$)

Let the significance level $\alpha = 0.05$

Under the null hypothesis, rounds of winning follow a binomial distribution with mean $7/2$, and variance $7/4$. Based on this, the probability that you win $6$ round or more is \begin{align} \binom{7}{6}\left(\frac{1}{2}\right)^{7} + \binom{7}{7}\left(\frac{1}{2}\right)^{7} \end{align}

Let's call this number $x$. Now, if the coin is unbiased, you win $6$ round or more out of 7 is very unlikely, so you reject your null hypothesis if $x$ is less than $\alpha$.

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