Question

Probability distributions: We pick six marbles one at a time out of a bag with sixty-six A. Hrosten distribution marbles of five different colors without putting them back in

Answer 1

The probability distribution for picking six marbles, one at a time, out of a bag with sixty-six A. Hrosten distribution marbles of five different colors without replacement follows a multinomial distribution.

Step-by-step explanation:

In this problem, we are dealing with a multinomial distribution since we are selecting marbles of different colors without replacement. The multinomial distribution is an extension of the binomial distribution and is used when there are more than two possible outcomes for each trial. In our case, the different colors of marbles represent the possible outcomes.

The formula for the probability mass function (PMF) of the multinomial distribution is given by:

`\[ P(X_1 = x_1, X_2 = x_2, ..., X_k = x_k) = (n!)/(x_1! \cdot x_2! \cdot ... \cdot x_k!) \cdot p_1^(x_1) \cdot p_2^(x_2) \cdot ... \cdot p_k^(x_k) \]`

Where:

- n is the total number of trials (in this case, the number of marbles picked, which is 6).

- k is the number of categories or colors of marbles.

-
`\(x_1, x_2, ..., x_k\)` are the counts for each category.

-
`\(p_1, p_2, ..., p_k\)` are the probabilities of selecting each category.

In our scenario, n = 6 and k = 5 (since there are five different colors). The probabilities
`\(p_1, p_2, ..., p_k\)` depend on the specific composition of marbles in the bag, which is not provided. If the bag has equal numbers of each color, then
`\(p_1 = p_2 = ... = p_k = (1)/(5)\)`.

Remember, without the specific probabilities for each color, a detailed calculation can't be provided, but the general approach using the multinomial distribution has been outlined.