2 variable binomial distribution?

[Master J]

I'm having a bit of trouble understanding a probability distribution of 2 variables.

Take for example taking n cards from a deck, and seeing what is the probability of getting X queens and say Y aces (with replacement). This involves the binomial distribution. The probabilities for the individual events are straight forward, but I'm having trouble getting the binomial coefficients.

Normally, for one variable, the coefficient is just n C X ( n choose X) where n is the total number of trials. This is n! / X! (n-X)! ... but how do you get this now for an X and a Y?

 

[Stephen Tashi]

I think you want a "multinomial distribution" and "multinomial coefficients" rather than a binomial distribution.

The binomial distribution can be understood from an analysis of the algebra used in computing the coefficients of terms in the expression $(p + q)^n$.

The multinomial distribution can be understood from analysing the algebra used in computing the coefficients of terms in expressions like $(p_a + p_q + p_s)^n$.

 

[Master J]

I don't know how searching for "multinomial" escaped me!

I know that the coefficient for my stated problem is as follows:

$\frac{n!}{X! Y! (n - X - Y)!}$

Does that seem correct? I haven't seen that particular form anywhere I've looked.

 

[Stephen Tashi]

That's correct for the coefficient. To get the probability, the coefficient is multiplied by a term of the form $p^X q^Y (1 - p - q)^{(n-X-Y)}$

 

[blue_raver22]

The binomial distribution is a commonly used probability distribution in statistics and probability theory. It is used to model the probability of obtaining a certain number of successes in a fixed number of trials, where each trial has only two possible outcomes (success or failure).

In the case of a 2 variable binomial distribution, we are interested in the probability of obtaining a specific combination of outcomes from two different variables, such as X queens and Y aces in a deck of cards. This involves using the binomial coefficient, which is a mathematical formula used to calculate the number of ways to choose a specific number of items from a larger set.

In this case, the binomial coefficient for two variables can be calculated using the formula (n choose X)(n choose Y), where n is the total number of trials and X and Y are the specific number of successes we are interested in. This can also be written as n! / (X!(n-X)!)(Y!(n-Y)!).

So, for example, if we wanted to calculate the probability of obtaining 2 queens and 3 aces in a deck of 10 cards, the binomial coefficient would be (10 choose 2)(10 choose 3) = 45 * 120 = 5400. This means that there are 5400 different ways to choose 2 queens and 3 aces from a deck of 10 cards.

In summary, the binomial distribution can be extended to include multiple variables by using the binomial coefficient formula for each variable. This allows us to calculate the probability of obtaining a specific combination of outcomes from multiple variables in a given number of trials.