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The premier league soccer team Chelsea always has a $\frac6{11}$ chance of winning a match.

Chelsea plays $6$ matches. What's the probability they win more than they lose?

I attempted this and thought that because Chelsea must win all $6$ games, or $5$, or $4$, the probability would be:

$$ \left(\frac6{11}\right)^6 + \left(\frac6{11}\right)^5 + \left(\frac6{11}\right)^4 $$

But this gives me $\approx 0.163$ which is far too small, as I'd expect the probability to be larger than $0.5$.

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    When winning 4 Matches, you also have to loose 2. Also you could win first, second, fifth and sixth Match. Or you could win second, third, fourth and sixth match... You have to add these up as well. – 2017-01-16
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    Why has this been downvoted? – 2017-02-21

2 Answers 2

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Split it into disjoint events, and then add up their probabilities:

  • The probability of exactly $\color\red4$ wins: $\binom{6}{\color\red4}\cdot\left(\frac{6}{11}\right)^{\color\red4}\cdot\left(1-\frac{6}{11}\right)^{6-\color\red4}$
  • The probability of exactly $\color\red5$ wins: $\binom{6}{\color\red5}\cdot\left(\frac{6}{11}\right)^{\color\red5}\cdot\left(1-\frac{6}{11}\right)^{6-\color\red5}$
  • The probability of exactly $\color\red6$ wins: $\binom{6}{\color\red6}\cdot\left(\frac{6}{11}\right)^{\color\red6}\cdot\left(1-\frac{6}{11}\right)^{6-\color\red6}$

Hence the overall probability is:

$$\sum\limits_{n=4}^{6}\binom{6}{n}\cdot\left(\frac{6}{11}\right)^{n}\cdot\left(1-\frac{6}{11}\right)^{6-n}$$

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    Why the combination? – 2017-01-16
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    @theonlygusti: Choosing the winning matches. – 2017-01-16
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    @theonlygusti Basically, you are dealing with a [Binomial Distribution](https://en.wikipedia.org/wiki/Binomial_distribution). – 2017-01-17
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Correct answer would be:

$$\left(\frac6{11}\right)^6+\left(\frac6{11}\right)^5\left(\frac5{11}\right){6\choose 5}+\left(\frac6{11}\right)^4\left(\frac5{11}\right)^2{6\choose 4}$$

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    Ok, I tried this but without the nCr product in each summand. Why are you multiplying each term by 6Cn? – 2017-01-16
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    To take all the cases into consideration, like WWWWLL,WWWLWL, etc. In other words, nCr is there to "choose the order of the losing games". – 2017-01-16