Use of probability in multiple choice questions. Don't have a mathematics background, please help.

[Wile_E_Coyote]

In an exam there are 100 questions. Each question has 4 options among which only one is correct. If a student marks the same options in all the questions then,
1) What is the probability of getting 20 or less than 20 correct answers out of 100?
2) What is the probability of getting 25 or more than 25 correct answers out of 100?
Which among the two is higher?

 

[Deleted member 4993]

In an exam there are 100 questions. Each question has 4 options among which only one is correct. If a student marks the same options in all the questions then,
1) What is the probability of getting 20 or less than 20 correct answers out of 100?
2) What is the probability of getting 25 or more than 25 correct answers out of 100?
Which among the two is higher?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

 

[Dr.Peterson]

In an exam there are 100 questions. Each question has 4 options among which only one is correct. If a student marks the same options in all the questions then,
1) What is the probability of getting 20 or less than 20 correct answers out of 100?
2) What is the probability of getting 25 or more than 25 correct answers out of 100?
Which among the two is higher?

In order to help effectively, we need to know what you know that might be useful, and in this case what tools are available. Do you know about the binomial distribution? Do you have software or tables that can give you such probabilities, or do you have to calculate it by hand?

 

[Wile_E_Coyote]

In order to help effectively, we need to know what you know that might be useful, and in this case what tools are available. Do you know about the binomial distribution? Do you have software or tables that can give you such probabilities, or do you have to calculate it by hand?

I always assumed that in a situation like this since the probability of randomly marking the correct answer is 25% and since the same option is marked everytime therefore we should get 25% of the questions correct. But when I tried this in real life, it rarely happened, to be honest it didn't happen at all, most of the times I got more than 25% correct, sometimes less than 25% and sometimes even 0% correct. You can explain whatever way you feel comfortable, understanding it will be my responsibility, because I asked the question and got the answer. Thank you.

 

[Wile_E_Coyote]

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

As I said, I don't have a mathematics background, so I did not come across this question in a textbook or some exercise. I am trying to make sense of a real life situation through this question. I always assumed that since the probability of getting a right answer is 25% and the same option is marked everytime, hence 25% questions should turn out to be correct, but in real life apparently that's not the case.

 

[Dr.Peterson]

With 100 questions, each having a 25% chance of being correct, the expected number of correct answers is 25. The probability of getting x questions right is a binomial distribution with p=.25, n=100. Using Excel, I get these probabilities for at least x questions correct:

x	P(X>=x)
0​	1​
5​	0.999999982​
10​	0.999956919​
15​	0.994579238​
20​	0.90046959​
25​	0.538328868​
30​	0.149541047​
35​	0.016426741​
40​	0.000686592​
45​	1.09214E-05​
50​	6.6385E-08​

So you'll get at least 25 right a little more than half the time; you'll get at least 20 right, 90% of the time (and fewer than 20, about 10% of the time).

You'll get exactly 25 right, 9% of the time.

 

[Steven G]

You should know that the chances of getting at least 20 correct will be greater than the chances of getting at least 25 correct. Do you see why?

 

[Wile_E_Coyote]

T

With 100 questions, each having a 25% chance of being correct, the expected number of correct answers is 25. The probability of getting x questions right is a binomial distribution with p=.25, n=100. Using Excel, I get these probabilities for at least x questions correct:

x	P(X>=x)
0​	1​
5​	0.999999982​
10​	0.999956919​
15​	0.994579238​
20​	0.90046959​
25​	0.538328868​
30​	0.149541047​
35​	0.016426741​
40​	0.000686592​
45​	1.09214E-05​
50​	6.6385E-08​

So you'll get at least 25 right a little more than half the time; you'll get at least 20 right, 90% of the time (and fewer than 20, about 10% of the time).

You'll get exactly 25 right, 9% of the

With 100 questions, each having a 25% chance of being correct, the expected number of correct answers is 25. The probability of getting x questions right is a binomial distribution with p=.25, n=100. Using Excel, I get these probabilities for at least x questions correct:

x	P(X>=x)
0​	1​
5​	0.999999982​
10​	0.999956919​
15​	0.994579238​
20​	0.90046959​
25​	0.538328868​
30​	0.149541047​
35​	0.016426741​
40​	0.000686592​
45​	1.09214E-05​
50​	6.6385E-08​

So you'll get at least 25 right a little more than half the time; you'll get at least 20 right, 90% of the time (and fewer than 20, about 10% of the time).

You'll get exactly 25 right, 9% of the time.

View attachment 32406 View attachment 32407

Thank you very much for your help, the graphs make it very easy to understand.