Solving Multinomial Dist Prob w 10 Dice: 2 & 4 Occur 3x Each

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In summary, the problem asks for the probability of getting 3 occurrences of the numbers 2 and 4 each when throwing 10 standard dice. The suggested solution involves calculating the probabilities for different combinations of these numbers appearing on the dice and adding them together. However, the correct answer is given as 0.0178, which differs from the solution provided. The discrepancy might be due to confusion about the number of combinations for each option.
  • #1
dfraser
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I'm struggling with this problem:

Suppose we throw 10 standard dice. Find the probability that faces 2 and 4 occur 3 times each.

I think the solution should look something like this:
$$4*{10\choose3,3,4}(1/6)^3(1/6)^3(1/6)^4 + 4*{10\choose3,3,3,1}(1/6)^3(1/6)^3(1/6)^3(1/6)^1$$
$$+ 6*{10\choose3,3,2,2}(1/6)^3(1/6)^3(1/6)^2(1/6)^2 + {10\choose3,3,1,1,1,1}(1/6)^3(1/6)^3(1/6)^1(1/6)^1(1/6)^1(1/6)^1$$

In particular, I reason that there are four ways that we can use up the remaining 4 faces: (1) they can all be the same (2) three can be the same and one different (3) we can have two pairs, and (4) all the faces can be different.

(1) can happen in 4 ways, (2) can happen in 4 ways, (3) can happen in 6 ways, and (4) can happen in 1 way. Hence my solution.

However, I'm told that the correct answer corresponds to .0178, which differs from my solution. I think I might be getting confused about the number of combinations for each option; can anyone give me some pointers?
 
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  • #2
Initially choose $3$ from $10$ positions for the $2$s.
Then choose from the remaining $7$ positions $3$ ones for the $4$s.
Fill the remaining $4$ positions with $1,3,5,6$.
 
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FAQ: Solving Multinomial Dist Prob w 10 Dice: 2 & 4 Occur 3x Each

What is a multinomial distribution?

A multinomial distribution is a probability distribution that describes the outcomes of a categorical variable with more than two categories. It is a generalization of the binomial distribution, which describes the outcomes of a binary variable.

How do you solve a multinomial distribution problem?

The general formula for solving a multinomial distribution problem is: P(X1=x1 and X2=x2 and ... and Xn=xn) = (n! / x1!x2!...xn!) * p1^x1 * p2^x2 * ... * pn^xn, where X1, X2, ..., Xn are the number of occurrences of each category, p1, p2, ..., pn are the probabilities of each category, and n is the total number of trials.

What does it mean when 2 and 4 occur 3 times each in a multinomial distribution?

In a multinomial distribution, the number of occurrences of each category is determined by the values of x1, x2, ..., xn. In this case, it means that out of 10 trials, the category with value 2 occurred 3 times and the category with value 4 also occurred 3 times.

How is the probability of a specific outcome calculated in a multinomial distribution?

The probability of a specific outcome in a multinomial distribution is calculated using the general formula mentioned above, where the values of X1, X2, ..., Xn correspond to the specific outcome and the probabilities p1, p2, ..., pn are known.

What are some real-life applications of multinomial distribution?

Multinomial distribution has many applications in various fields, such as market research, election prediction, genetics, and sports analytics. For example, it can be used to analyze the results of a survey with multiple response options or to predict the outcome of a multi-party election.

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