Question

Nts) in a certain population of voters, 72% of them plan to vote in favor of a new

piece of proposed legislation. If a random sample of 25 people is taken from this
population, find the probability that 10 of them plan to vote against this piece of
proposed legislation.​

Answer 1

Probability that 10 of them plan to vote against this piece of proposed legislation is 0.0701 or 7.01 %.

Explanation:

Consider the event of voting against the piece as success, 'p'. So, voting in favor of the piece is failure and denoted by 'q'.

Given:

Sample size is,
`n=25`

Probability of failure is,
`q=72\%=0.72`

Therefore, probability of success is,
`p=1-q=1-0.72=0.28`

Number of successes is,
`x=10`

Now, from Bernoulli's distribution, probability of
`x` successes out of
`n` samples is given as:

`P(X=x)=_(n)^(x)\textrm{C}p^xq^(n-x)`

Here,
`n=25,x=10,p=0.28,q=0.72`. Therefore,

`P(X=10)=_(25)^(10)\textrm{C}(0.28)^(10)(0.72)^(25-10)\\P(X=10)=_(25)^(10)\textrm{C}(0.28)^(10)(0.72)^(15)\\P(X=10)=3.269* 10^(6)* 2.962* 10^(-6)* 7.244* 10^(-3)\\P(X=10)=70.142* 10^(-3)=0.0701=7.01\%`

Therefore, probability that 10 of them plan to vote against this piece of proposed legislation is 0.0701 or 7.01 %.