MATHEMATICAL EXPLORATION MULTINOMIAL DISTRIBUTION
Rationale
I chose to research the multinomial distribution because I enjoyed learning and applying the binomial distribution to real life examples in order to find the discrete probability distribution of the number of successes in a sequence of independent experiments. The multinomial distribution is a generalisation of the binomial distribution. The binomial distribution only works where there are only two possible outcomes, therefore I wanted to explore how to model the probability of the number of counts for more than two possibilities. For example, the binomial distribution can be used to model the probability of getting a given number of heads while flipping a coin a certain number of times, but what if you wanted to find out the probability distribution of a dice for large numbers of trials? This uses the multinomial distribution, because there are more than 6 possible outcomes in each trial rather than just 2.
Explanation of the multinomial distribution
Suppose:
• There are independent trials.
• Each trial results in one of mutually exclusive outcomes
• On any single trial, these outcomes occur with probabilities
• The probabilities stay constant from trial to trial.
Let the random variable represent the number of occurrences of outcome .
Then the probability mass function is:
Derivation of the multinomial distribution
The probability of occurring in any trial is , therefore the probability of occurring times is equal to . Therefore, the probability of any specific ordering of occurrences of outcome 1 and occurrences of outcome 2 all the way up to occurrences of outcome is:
We need to know how many possible orderings there are for . Using permutations, there are possible orderings of different values, but since can take any value up to , the number of arrangements is
We now know the probability of any single arrangement of occurrences of outcome 1 all the way up to occurrences of outcome , and the number of possible orderings, therefore the probability mass function for random variables occurring times in trials is:
Any one random variable Xi when viewed individually will have a binomial distribution, since so probability of the other variables occurring will have .
Using this, we can find the expected value and variance for .
Example 1: Rolling a dice
One example of a multinomial distribution is the probability of rolling a given number of each dice face in a given number of trials. The roll X can take the values 1,2,3,4,5 and 6. This is the probability mass function where x is the number on the dice:
x
1
2
3
4
5
6
P(X=x)
Using the binomial distribution, we can find the probability of rolling any number of ones in 6 trials:
This is the probability mass function where x is equal to the number of 1s rolled in 6 trials
0
1
2
3
4
5
6
0.335
0.402
0.201
0.0536
0.00804
0.000643
0.0000214
The binomial distribution can model the probability when there are two possible outcomes, but the multinomial is needed when there are more than 2. For example, what is the probability that each number is rolled once in 6 trials? Using the formula:
While the probability of rolling any number exactly once in 6 trials is 0.402, the probability of rolling one of each number in 6 trials is much smaller at 0.0154.
Given that rolling each number has a probability of , a fair sided die will show each number times for a large number of trials if it is truly random. The multinomial distribution, however, is useful to show that when is small, it is actually extremely unlikely for the dice to show each number an equal number of times.
Example 2: Selecting random voters
Outcome
Party
Vote share % (0 decimal places)
Probability of selection
X1
Conservative
42
0.42
X2
Labour
40
0.40
X3
Other
18
0.18
Suppose 100 people who voted in the UK 2017 General Election were randomly selected to answer a survey on who they voted for. Using the multinomial distribution, what is the probability that the number of voters for each party is the same to zero decimal places as the percentage vote share of each party?
Therefore, there is a less than one thousandth probability that a survey of just one hundred people will give an accurate result for the survey. The multinomial distribution can be used to show that problems with surveys of a small sample size.
Example 3: Balls in an urn
The multinomial distribution can be used to find the probability of given numbers of coloured balls removed from an urn with replacement, and this method can be applied to other scenarios.
(i) An urn contains 6 red balls, 5 yellow balls, 9 white balls. 5 balls are randomly selected with replacement
What is the probability 1 is red, 2 are yellow and 3 are white?
If we are putting the balls back in before the next trial, the individual trials are independent, and the probabilities are staying constant.
= number of red balls, = number of yellow balls, = number of white balls selected.
There are 20 balls in total, so the probabilities of selecting the colours in each trial are
This problem can be extended to sampling without replacement. The number of each balls and the total number of balls will change, therefore the trials are no longer independent so the conditions of the multinomial distribution will no longer be satisfied. We have to use the multivariate hypergeometric distribution.
(ii) An urn contains 6 red balls, 5 yellow balls, 9 white balls. 5 balls are randomly selected without replacement.
What is the probability 1 is red, 2 are yellow and 3 are white?
We still want to find
The total number of combinations that can be selected is
The number of combinations that have 1 red ball, 2 yellow balls and 3 white balls is .
Notice that the probability with replacement is different to the probability without replacement. The multinomial distribution cannot be used for some multivariate distributions, for example in dealing cards or choosing people from a small sample.
Conclusion
In this exploration, my aim was to understand the multinomial distribution to further my understanding of statistics and probability, and answer problems by applying the multinomial distribution formula. Since the probability of rolling any one side on a die is , a common (but nevertheless incorrect) misconception is that if you roll it six times, each side will come up once. However, I found using the multinomial distribution that the probability of this occurring is only 0.0154, which is fairly unlikely.
While the multinomial distribution has applications when the probabilities remain constant from trial to trial, in many real-life examples this is not the case. I briefly explored an example of the multivariate hypergeometric distribution, and I after this exploration I have been left with a desire to explore more distributions that are not covered in the IB course and apply them to real-life situations.
Bibliography
Balka, J., 2012. Introduction to the Multinomial Distribution. [Online]
Available at: http://www.jbstatistics.com
House of Commons Library, 2017. General Election 2017: results and analysis. [Online]
Available at: http://www.parliament.uk/commons-library
International Baccalaureate Organisation, 2012. Mathematics HL and further mathematics HL formula booklet. s.l.:s.n.