# Probability of a majority decision

There is a 1001 man jury and each member has a (2/3) probability of making the correct decision. What is the probability that the majority decision is correct?

I want to say that the answer is 1 - [(1/3)(500/1001) + (2/3)(501/1001)] = .5 but I'm not sure. Thanks!

The majority is correct, if 501 to 1001 of the 1001 man are correct.

You have to use the cdf of the binomial distribution:

$P(X \geq 501)=\sum_{x=501}^{1000}{1000 \choose x}\cdot \left(\frac{2}{3}\right)^x\cdot\left(\frac{1}{3}\right)^{1000-x}$

Applying converse probability

$P(X \geq 501)=1-P(X \leq 500)=1-\sum_{x=0}^{500}{1000 \choose x}\cdot \left(\frac{2}{3}\right)^x\cdot\left(\frac{1}{3}\right)^{1000-x}$

Applying Moivre-Laplace-Theorem

$$P(X \geq 501)=1-\Phi\left(\frac{500-2/3\cdot 1000}{\sqrt{1000\cdot 2/3 \cdot 1/3}} \right)$$

• I have downvoted. There are two errors here. (1) There should be "1000" rather than "100" in the numerator of the fraction. (2) A continuity corection should be applied because you are using a continuous random variable to approximate a discrete random variable, so the "500" should be "500.5" – tomi May 7 '15 at 21:58
• Historical note: Even twenty years ago the calculation of $P(\leq500)$ would have been prohibitive, which is why approximations were used. With modern technology it has become unnecessary to use an approximating Normal distribution in a case like this; Excel (and other spreadsheets) will happily calculate these for you. As the approximation is not as accurate, you are better to stick to the exact distribution where possible. – tomi May 7 '15 at 22:03
• For this problem Excel says the true value is $1-8.845\times 10^{-28}$ but says the approximation is $1-2.389\times 10^{-29}$ – tomi May 7 '15 at 22:05
• @tomi a continous correction factor is not neccessary. n is very large. Should I downvote your comment ? – callculus May 7 '15 at 22:09
• I have removed my downvote and have now upvoted your corrected answer. – tomi May 7 '15 at 22:34

If X is the number of jurors to make a correct decision, then $X$ has a Binomial distribution $B\left(1001,\frac 2 3\right)$.

You need to find $P\left(X\geq 501\right)= 1-P\left(X\leq500\right)$

You can get this from Excel and the result is very very very close to 1.

With a jury of 12, the probability of a correct decision is 0.8223