Help with finite math!!!

Question

Suppose that a pizzeria has 12 pizzas on their menu: 7 of them include meat and the rest are vegetarian. Five friends go there and each randomly orders one from the menu. What is the probability that they will get at least 4 vegetarian pizzas? (Leave answer as an expression)!!!!

Answer 1

Probability of getting at least 4 vegie pizza means getting exactly 4 veg or exactly 5 veg pizzas. Getting exactly 4 veg pizzas require you to get the following:

 

1. Choose 4 friends from 5 people. There are 5choose4 ways to do so, also written as nchoosek(5,4). This equals to 5.

2. For each of those combinations, you need to have the 4 people that got selected, gets a veg pizza each. Getting one of those 7 veg pizza out of total of 12 means a probability of (7/12). Since this probability gets repeated for four of them, multiple this number four times: (7/12)⁴

3. Finally, the remaining friend has to choose a meat pizza. The probability is: (5/12)¹

 

Thus getting exactly 4 veg pizza has a probability:

P(4) = nchoosek(5, 4) * (7/12)⁴ * (5/12)¹

 

Similarly, P(5) = nchoosek(5,5) * (7/12)⁵ * (5/12)⁰

 

So finally just add these two up: P(4)+P(5), and you get your answer.

 

Now if you understood the construction of this discussion, you'll see in your textbook that it talks about a technique called binomial formula / theorem. This problem is just an example of that. The binomial formula gives you a general technique to solve these types of problems.