Question

A survey of eating habits showed that approximately 44​% of people in a certain city are vegans. vegans do not eat​ meat, poultry,​ fish, seafood,​ eggs, or milk. a restaurant in the city expects 350350 people on opening​ night, and the chef is deciding on the menu. treat the patrons as a simple random sample from the city and the surrounding​ area, which has a population of about​ 600,000. if 1414 vegan meals are​ available, what is the approximate probability that there will not be enough vegan mealslong dash—that ​is, the probability that 1515 or more vegans will come to the​ restaurant? assume the vegans are independent and there are no families of vegans.

Answer 1

Given a binomial distribution, the probability can be approximated using a normal distribution as follows:

μ = np = 350(0.04) = 14
$\sigma=\sqrt{np(1-p)} \\ \\ =\sqrt{350(0.04)(0.96)} \\ \\ =\sqrt{13.44}=3.666$

$P(x\geq15)=1-P(x\ \textless \ 15) \\ \\ =1-P\left(z\ \textless \ \frac{15-14}{3.666/\sqrt{350}} \right)=1-P\left(z\ \textless \ \frac{1}{0.196} \right) \\ \\ =1-P(5.1032)=1-1=0$

Therefore, the probability that there will not be enough vegan meals is 0.