The Notes: In 2014 it was reported that 72% of Australians supported marriage equality. Find the probability that more than 12 out of a randomly selected...

Wednesday, November 27, 2013

In 2014 it was reported that 72% of Australians supported marriage equality. Find the probability that more than 12 out of a randomly selected...

Whenever we need to determine the probability of multiple true/false outcomes (support marriage equality or don't support marriage equality) out of some predetermined number of Bernoulli trials, we are dealing with binomial distribution.

Probability mass function of binomial random variable is given by

$\displaystyle{\left(\matrix{{n}\\{k}}\right)}{{p}}^{{k}}{{\left({1}-{p}\right)}}^{{{n}-{k}}}$

where $\displaystyle{n}$ is the number of Bernoulli trials, $\displaystyle{k}$ is number of successful (true) trials and $\displaystyle{p}$ is the probability of success.

In this case

$\displaystyle{n}={20}$

$\displaystyle{k}>{12}$

$\displaystyle{p}={72}%={0.72}$

Therefore, the probability that more than 12 Australians out of 20 support marriage equality will be the sum of the mass function for each $\displaystyle{k}$ between 13 and 20 i.e.

$\displaystyle{\sum_{{{k}={13}}}^{{20}}}{\left(\matrix{{20}\\{k}}\right)}{{0.72}}^{{k}}{{0.28}}^{{{20}-{k}}}$

For $\displaystyle{k}={13}$ we get

$\displaystyle{\left(\matrix{{20}\\{13}}\right)}{{0.72}}^{{13}}{{0.28}}^{{7}}=\frac{{{20}\cdot{19}\cdot{18}\cdot{17}\cdot{16}\cdot{15}\cdot{14}\cdot}}{{{1}\cdot{2}\cdot{3}\cdot{4}\cdot{5}\cdot{6}\cdot{7}}}\cdot{{0.72}}^{{13}}{{0.28}}^{{7}}\approx{0.146165}$

In the line above we have used the fact that $\displaystyle{\left(\matrix{{n}\\{k}}\right)}={\left(\matrix{{n}\\{n}-{k}}\right)}$ which can be useful in calculating binomial coefficients.

We proceed by calculating the terms of the sum for all $\displaystyle{k}={13},{14},\ldots,{20}$ and then summing all the terms.

Finally we get

$sum_(k=13)^20 ((20),(k))0.72^k0.28^(20-k)\approx0.829272=82.9272%$

Therefore, we can conclude that there is $\displaystyle{82.9272}%$ probability that more than 12 randomly selected Australians in 2014 supported marriage equality.