A study determined that 15% of a population use a certain brand of laundry detergent. What is the probability that more than 2 shoppers selected at random from 12 shoppers use that brand of detergent?

Question

A study determined that 15% of a population use a certain brand of laundry detergent.  What is the probability that more than 2 shoppers selected at random from 12 shoppers use that brand of detergent?

kirbykirby · Answer

You can recognize this being a binomial problem. If you let $X$ be the number of shoppers selected that use the brand of detergent, then $X$ follows a binomial distribution.

The question asks for $P(X>2)$, which is the same as $P(X \ge 3)$ since the random variable is discrete. This is a bit long to calculate, so it's easier to calculate this with the aid of the complement, and find $1-P(X \le 2)$. 

$$P(X \le 2)= P(X=0)+P(X=1)+P(X=2)$$
and you can find $P(X = x)$ as $$\large {12 \choose x} 0.15^x(1-0.15)^{12-x}$$, replacing $x$ with 0, 1 and 2

anonymous · Answer

ive tried plugging it into my calculator as a binomial but i keep getting the wrong answer

kirbykirby · Answer

$$ P(X \le 2)= {12 \choose 0} 0.15^0(0.85)^{12}+ {12 \choose 1} 0.15^1(0.85)^{11}+{12 \choose 2} 0.15^2(0.85)^{10} \=0.7358$$

kirbykirby · Answer

?

anonymous · Answer

i keep getting .2924 with my calculator but i know that isnt right, i think yours is!

kirbykirby · Answer

And then since we needed to use the complement, 1 - 0.7358 = 0.2642

anonymous · Answer

so .2642 is the final?

kirbykirby · Answer

Yes, should be. Do they give you an answer?

anonymous · Answer

no I wish but I'm sure you're right!

kirbykirby · Answer

I'm fairly certain about my answer, but to be honest I am quite tired at this time so maybe I made a mistake and don't realize it o.o!

anonymous · Answer

better than what i have and i know mine isn;t right! haha thank you Kirby!

kirbykirby · Answer

your welcome! :)