# Is It Hypergeometric or Binomial: Calculating Guessing Probabilities on Exams?

• Somefantastik
In summary, the probability of a student getting four or more correct answers on a multiple choice exam with 3 possible choices for each of the 5 questions is (^{5}_{4})p^{4}q + (^{5}_{5})p^{5} where p = 1/3 and q = 2/3. This is based on the binomial distribution since each question has a binary outcome - either the student chooses the correct answer or not.
Somefantastik
On a multiple guess exam, there are 3 possible answers for each of the 5 questions. What is the probability that the student will get four or more correct answers just by guessing?

Is this hypergeometric or binomial?

Since on each question a student can either choose the correct answer or not, it is binomial. Do you see what is the probability a student will choose the correct answer on a specific question just by guessing?

I thought it was either this

$$\frac{(^{5}_{4}) (^{10}_{1})}{(^{15}_{5})}$$ choose 4 from 5 correct answers, choose 1 from 10 wrong answers divided by choosing 5 answers from 15

Or this (which I now know is probably the right answer)

$$(^{5}_{4})p^{4}q + (^{5}_{5})p^{5}$$ which is the probability of 4/5 right answers + probability of 5/5 right answers, where prob correct answer = 1/3

Yes, the second is the one you want (you are NOT choosing 4 or 5 correct answers from all 15- that would imply that you could choose 2 correct answer from one problem and none from another!). Now what number is $$(^{5}_{4})p^{4}q + (^{5}_{5})p^{5}$$?

p = 1/3 and q = 2/3 => 0.14 unless I calculated wrong which is entirely possible.

## 1. What is a probability distribution?

A probability distribution is a mathematical function that describes the likelihood of a random variable taking on a certain value or set of values. It shows the possible outcomes of an event and the probability of each outcome occurring.

## 2. What are the types of probability distributions?

There are several types of probability distributions, including the normal distribution, binomial distribution, Poisson distribution, and exponential distribution. Each type is used to model different types of data and has its own unique characteristics.

## 3. How is a probability distribution different from a histogram?

A histogram is a graphical representation of data, while a probability distribution is a mathematical function. A histogram shows the frequency of values in a dataset, while a probability distribution shows the probability of each value occurring.

## 4. How do you calculate the mean and variance of a probability distribution?

The mean of a probability distribution is calculated by multiplying each possible value by its probability and summing up all the products. The variance is calculated by taking the difference between each value and the mean, squaring it, multiplying by its probability, and summing up all the products.

## 5. How is a probability distribution used in real life?

Probability distributions are used in a variety of fields, including finance, economics, psychology, and physics. They are used to model and analyze data, make predictions, and make decisions based on uncertainty. For example, they can be used to calculate the risk of a stock portfolio or the likelihood of a medical treatment being effective.

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