Probability: binomial coefficient problem

• Jaysquared
In summary, the conversation discusses a probability problem involving three restaurants and nine customers. The question is to find the probability that three customers will go to each restaurant. The use of binomial coefficients and the multinomial distribution are suggested for solving the problem.

Jaysquared

Hello physics forum! I come to you with a binomial coefficient problem I am stuck on. Here is the question

1. Suppose an airport has three restaurants open, Subway Burger King and McDonalds. If all three restaurants are open and each customer is equally likely to go to each one, what is the probability that out of 9 customers, three will go to each one?

2. Permutations and binomial coefficient comes in handy for this

3. Alright so my initial idea is that since all three restaurants are open, there is a 1/3 chance of choosing one of them. Since there are 9 people choosing one of the three restaurants, It would be (1/3)^9. Using the binomial coefficient the initial setup would be 9 choose 3. Am I on the right track? The answer that they give is 0.085. Please help for any suggestions. Thank you!

Let's introduce the following notation:
##A = \{## exactly 3 people dine at Subway ##\}##
##B = \{## exactly 3 people dine at BK ##\}##
##C = \{## exactly 3 people dine at McD ##\}##
Then you are trying to calculate ##P(A \cap B \cap C)##. This is the same as ##P(A \cap B)## (why?). Try calculating it using ##P(A \cap B) = P(A | B)P(B)##.

jbunniii said:
Let's introduce the following notation:
##A = \{## exactly 3 people dine at Subway ##\}##
##B = \{## exactly 3 people dine at BK ##\}##
##C = \{## exactly 3 people dine at McD ##\}##
Then you are trying to calculate ##P(A \cap B \cap C)##. This is the same as ##P(A \cap B)## (why?). Try calculating it using ##P(A \cap B) = P(A | B)P(B)##.

Alternatively: use the multinomial distribution (trinomial in this case); see, eg., http://en.wikipedia.org/wiki/Multinomial_distribution or http://mathworld.wolfram.com/MultinomialDistribution.html

What is a binomial coefficient?

A binomial coefficient is a mathematical term used to represent the number of ways to choose a specified number of objects from a larger set. It is also known as a combination.

How is a binomial coefficient calculated?

The binomial coefficient is calculated using the formula nCr = n!/(r!(n-r)!), where n represents the total number of objects, and r represents the number of objects being chosen.

What is the significance of binomial coefficients in probability?

In probability, binomial coefficients are used to calculate the probability of a specific outcome in a series of events, such as flipping a coin multiple times or drawing cards from a deck. They help determine the likelihood of obtaining a certain number of successes in a given number of trials.

Can binomial coefficients be used for more than two outcomes?

Yes, binomial coefficients can be used for any number of outcomes. They are often used in situations where there are more than two possible outcomes, such as rolling a die or selecting from a group of candidates.

What are some real-life applications of binomial coefficients?

Binomial coefficients have various applications in fields such as statistics, genetics, and economics. They can be used to analyze and predict outcomes in sports games, elections, and market trends. In genetics, they can be used to determine the probability of inheriting certain traits. They are also commonly used in computer science and programming for tasks such as data compression and error-correcting codes.