# Determining probability using combinations/permutations (i think)

• subopolois
In summary, the probability of a student getting a perfect score on the final exam is 0.1087, assuming they only know how to solve 25 out of 30 questions. For getting at least 8 questions right, the student can calculate the probability of getting 8, 9, or 10 questions right and add it to the probability from the first question. The student can use permutations and combinations to find these probabilities.
subopolois

## Homework Statement

a student has an exam, the teacher gives 30 questions and 10 of the 30 will be on the final exam. if the student knows how to solve 25 of the 30, what is:
a) the probability he will get perfect
b) the probability he will get at least 8 questions right

## Homework Equations

permutations and combinations

## The Attempt at a Solution

for part a i have:
sample space= 30Choose10= 30045015
to get all 10 questions right, the 10 questions on the exam must be within the 25 he knows how to do- 25choose10= 3268760. to find the probability i did: 3268760/30045015= 0.1087.
i don't think this is right because it seems too low and since he knows most of the questions i would expect the probability to be higher.

for part b: i use the same sample space of 30choose10, but I am stuck on the "at least 8 questions right part"

any help for this would be appreciated

You've got the first one right. Yeah, it seems kind of low, but then he only knows how to solve 'most of them'. And he has to get 10 right in a row. It's not that unreasonable. For the second one what's the probability he gets 9 questions right and 1 wrong and what's the probability he get 8 right and 2 wrong. Then add those two to the probability from the first question.

I would approach this problem using the concept of conditional probability. We know that the student knows how to solve 25 out of 30 questions, so the probability of getting a question right is 25/30 or 0.8333. Now, for part a, the probability of getting all 10 questions right would be (0.8333)^10 = 0.1945, which is significantly higher than the value you calculated.

For part b, we can use the concept of binomial distribution to calculate the probability. The formula for this is P(X=k) = nCk * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success. In this case, n=10, k=8, and p=0.8333. So the probability of getting at least 8 questions right would be P(X=8) + P(X=9) + P(X=10), which can be calculated using the formula. This gives a probability of 0.5723, which is significantly higher than the value you calculated.

In summary, using the concepts of conditional probability and binomial distribution, we can calculate more accurate probabilities for the student's performance on the exam.

## 1. How do combinations and permutations differ in determining probability?

Combinations and permutations are both methods of determining the probability of a certain outcome. Combinations are used when the order of elements does not matter, while permutations are used when the order does matter.

## 2. When should I use combinations instead of permutations?

Combinations should be used when the order of elements is irrelevant, such as when choosing a group of people for a committee or selecting a group of items from a larger set. Permutations should be used when the order of elements is important, such as in arranging a sequence of events.

## 3. How do I calculate the number of combinations for a given situation?

The formula for calculating combinations is nCr = n! / r!(n-r)!, where n is the total number of elements and r is the number of elements to be selected. The exclamation point represents the factorial function, which means multiplying a number by all the numbers below it.

## 4. Can I use combinations and permutations to determine the probability of winning a game of chance?

Yes, combinations and permutations can be used to determine the probability of winning a game of chance. For example, if there are 10 possible outcomes and you need to match 5 to win, you can use the combination formula to calculate the probability of winning.

## 5. Is there a practical application for using combinations and permutations in real life?

Yes, combinations and permutations are used in various fields such as statistics, probability, and computer science. They are used to solve problems involving combinations of objects, arranging elements in a specific order, and predicting outcomes in different scenarios.

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