Answer:
a 
b 
c 
Step-by-step explanation:
Given that:
In a multiple choice quiz:
there are 5 questions
and 4 choices for each question (a, b, c, d)
let X be the correctly answered question = 1 answer only
and Y be the choices for each question = 4 choices
The probability that Robin guessed the correct answer is:
Probability = n(X)/n(Y)
Probability = 1//4
Probability = 0.25
The probability mass function is :

We are to find the required probability that the first question she gets right is the 3rd question.
i.e when x = 3


b) Find the probability that She gets exactly 3 or exactly 4 questions right
we know that :
n = 5 questions
Probability P =0.25
Let represent X to be the number of questions guessed correctly i,e 3 or 4
Then; the probability mass function can be written as:
![\[P\left( {X = x} \right) = \left( {\begin{array}{*{20}{c}}\\5\\\\x\\\end{array}} \right){\left( {0.25} \right)^x}{\left( {1 - 0.25} \right)^{5 - x}}\]](https://tex.z-dn.net/?f=%5C%5BP%5Cleft%28%20%7BX%20%3D%20x%7D%20%5Cright%29%20%3D%20%5Cleft%28%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D%5C%5C5%5C%5C%5C%5Cx%5C%5C%5Cend%7Barray%7D%7D%20%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5Ex%7D%7B%5Cleft%28%20%7B1%20-%200.25%7D%20%5Cright%29%5E%7B5%20-%20x%7D%7D%5C%5D)

![\[P\left( {X = 3 \ or \ 4 } \right) = \left( {\begin{array}{*{20}{c}}\\5\\\\3\\\end{array}} \right){\left( {0.25} \right)^3}{\left( {1 - 0.25} \right)^{5 - 3}}\] + \left( {\begin{array}{*{20}{c}}\\5\\\\4\\\end{array}} \right){\left( {0.25} \right)^4}{\left( {1 - 0.25} \right)^{5 - 4}}\]](https://tex.z-dn.net/?f=%5C%5BP%5Cleft%28%20%7BX%20%3D%203%20%5C%20or%20%5C%204%20%7D%20%5Cright%29%20%3D%20%5Cleft%28%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D%5C%5C5%5C%5C%5C%5C3%5C%5C%5Cend%7Barray%7D%7D%20%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E3%7D%7B%5Cleft%28%20%7B1%20-%200.25%7D%20%5Cright%29%5E%7B5%20-%203%7D%7D%5C%5D%20%2B%20%20%5Cleft%28%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D%5C%5C5%5C%5C%5C%5C4%5C%5C%5Cend%7Barray%7D%7D%20%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E4%7D%7B%5Cleft%28%20%7B1%20-%200.25%7D%20%5Cright%29%5E%7B5%20-%204%7D%7D%5C%5D)
![\[P\left( {X = 3 \ or \ 4 } \right) = \dfrac{5!}{3!(5-3)!}\right){\left( {0.25} \right)^3}{\left( {1 - 0.25} \right)^{5 - 3}}\] + \dfrac{5!}{4!(5-4)!} \right){\left( {0.25} \right)^4}{\left( {1 - 0.25} \right)^{5 - 4}}\]](https://tex.z-dn.net/?f=%5C%5BP%5Cleft%28%20%7BX%20%3D%203%20%5C%20or%20%5C%204%20%7D%20%5Cright%29%20%3D%20%20%5Cdfrac%7B5%21%7D%7B3%21%285-3%29%21%7D%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E3%7D%7B%5Cleft%28%20%7B1%20-%200.25%7D%20%5Cright%29%5E%7B5%20-%203%7D%7D%5C%5D%20%2B%20%5Cdfrac%7B5%21%7D%7B4%21%285-4%29%21%7D%20%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E4%7D%7B%5Cleft%28%20%7B1%20-%200.25%7D%20%5Cright%29%5E%7B5%20-%204%7D%7D%5C%5D)
![\[P\left( {X = 3 \ or \ 4 } \right) = \dfrac{5!}{3!(2)!}\right){\left( {0.25} \right)^3}{\left( {0.75} \right)^{2}}\] + \dfrac{5!}{4!(1)!} \right){\left( {0.25} \right)^4}{\left( {0.75} \right)^{1}}\]](https://tex.z-dn.net/?f=%5C%5BP%5Cleft%28%20%7BX%20%3D%203%20%5C%20or%20%5C%204%20%7D%20%5Cright%29%20%3D%20%20%5Cdfrac%7B5%21%7D%7B3%21%282%29%21%7D%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E3%7D%7B%5Cleft%28%20%7B0.75%7D%20%5Cright%29%5E%7B2%7D%7D%5C%5D%20%2B%20%5Cdfrac%7B5%21%7D%7B4%21%281%29%21%7D%20%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E4%7D%7B%5Cleft%28%20%7B0.75%7D%20%5Cright%29%5E%7B1%7D%7D%5C%5D)


c) Find the probability if She gets the majority of the questions right.
We know that the probability mass function is :
![\[P\left( {X = x} \right) = \left( {\begin{array}{*{20}{c}}\\5\\\\x\\\end{array}} \right){\left( {0.25} \right)^x}{\left( {1 - 0.25} \right)^{5 - x}}\]](https://tex.z-dn.net/?f=%5C%5BP%5Cleft%28%20%7BX%20%3D%20x%7D%20%5Cright%29%20%3D%20%5Cleft%28%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D%5C%5C5%5C%5C%5C%5Cx%5C%5C%5Cend%7Barray%7D%7D%20%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5Ex%7D%7B%5Cleft%28%20%7B1%20-%200.25%7D%20%5Cright%29%5E%7B5%20-%20x%7D%7D%5C%5D)
So; of She gets majority of her answers right ; we have:
The required probability is,

∴
![\[P\left( {X = 3 \ or \ 4 \ or \ 5 } \right) = \left( {\begin{array}{*{20}{c}}\\5\\\\3\\\end{array}} \right){\left( {0.25} \right)^3}{\left( {1 - 0.25} \right)^{5 - 3}}\] + \left( {\begin{array}{*{20}{c}}\\5\\\\4\\\end{array}} \right){\left( {0.25} \right)^4}{\left( {1 - 0.25} \right)^{5 - 4}}\]+ \left( {\begin{array}{*{20}{c}}\\5\\\\5\\\end{array}} \right){\left( {0.25} \right)^5}{\left( {1 - 0.25} \right)^{5 - 5}}\]](https://tex.z-dn.net/?f=%5C%5BP%5Cleft%28%20%7BX%20%3D%203%20%5C%20or%20%5C%204%20%5C%20or%20%20%5C%205%20%7D%20%5Cright%29%20%3D%20%5Cleft%28%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D%5C%5C5%5C%5C%5C%5C3%5C%5C%5Cend%7Barray%7D%7D%20%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E3%7D%7B%5Cleft%28%20%7B1%20-%200.25%7D%20%5Cright%29%5E%7B5%20-%203%7D%7D%5C%5D%20%2B%20%20%5Cleft%28%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D%5C%5C5%5C%5C%5C%5C4%5C%5C%5Cend%7Barray%7D%7D%20%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E4%7D%7B%5Cleft%28%20%7B1%20-%200.25%7D%20%5Cright%29%5E%7B5%20-%204%7D%7D%5C%5D%2B%20%20%5Cleft%28%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D%5C%5C5%5C%5C%5C%5C5%5C%5C%5Cend%7Barray%7D%7D%20%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E5%7D%7B%5Cleft%28%20%7B1%20-%200.25%7D%20%5Cright%29%5E%7B5%20-%205%7D%7D%5C%5D)
![\[P\left( {X = 3 \ or \ 4 \ or \ 5 } \right) = \dfrac{5!}{3!(5-3)!}\right){\left( {0.25} \right)^3}{\left( {1 - 0.25} \right)^{5 - 3}}\] + \dfrac{5!}{4!(5-4)!} \right){\left( {0.25} \right)^4}{\left( {1 - 0.25} \right)^{5 - 4}}\] + \dfrac{5!}{5!(5-5)!} \right){\left( {0.25} \right)^5}{\left( {1 - 0.25} \right)^{5 - 5}}\]](https://tex.z-dn.net/?f=%5C%5BP%5Cleft%28%20%7BX%20%3D%203%20%5C%20or%20%5C%204%20%5C%20or%20%5C%205%20%7D%20%5Cright%29%20%3D%20%20%5Cdfrac%7B5%21%7D%7B3%21%285-3%29%21%7D%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E3%7D%7B%5Cleft%28%20%7B1%20-%200.25%7D%20%5Cright%29%5E%7B5%20-%203%7D%7D%5C%5D%20%2B%20%5Cdfrac%7B5%21%7D%7B4%21%285-4%29%21%7D%20%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E4%7D%7B%5Cleft%28%20%7B1%20-%200.25%7D%20%5Cright%29%5E%7B5%20-%204%7D%7D%5C%5D%20%2B%20%5Cdfrac%7B5%21%7D%7B5%21%285-5%29%21%7D%20%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E5%7D%7B%5Cleft%28%20%7B1%20-%200.25%7D%20%5Cright%29%5E%7B5%20-%205%7D%7D%5C%5D)
![\[P\left( {X = 3 \ or \ 4 \ or \ 5 } \right) = \dfrac{5!}{3!(5-3)!}\right){\left( {0.25} \right)^3}{\left( {1 - 0.75} \right)^{2}}\] + \dfrac{5!}{4!(5-4)!} \right){\left( {0.25} \right)^4}{\left( {0.75} \right)^{1}}\] + \dfrac{5!}{5!(5-5)!} \right){\left( {0.25} \right)^5}{\left( {0.75} \right)^{0}}\]](https://tex.z-dn.net/?f=%5C%5BP%5Cleft%28%20%7BX%20%3D%203%20%5C%20or%20%5C%204%20%5C%20or%20%5C%205%20%7D%20%5Cright%29%20%3D%20%20%5Cdfrac%7B5%21%7D%7B3%21%285-3%29%21%7D%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E3%7D%7B%5Cleft%28%20%7B1%20-%200.75%7D%20%5Cright%29%5E%7B2%7D%7D%5C%5D%20%2B%20%5Cdfrac%7B5%21%7D%7B4%21%285-4%29%21%7D%20%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E4%7D%7B%5Cleft%28%20%7B0.75%7D%20%5Cright%29%5E%7B1%7D%7D%5C%5D%20%2B%20%5Cdfrac%7B5%21%7D%7B5%21%285-5%29%21%7D%20%5Cright%29%7B%5Cleft%28%20%7B0.25%7D%20%5Cright%29%5E5%7D%7B%5Cleft%28%20%7B0.75%7D%20%5Cright%29%5E%7B0%7D%7D%5C%5D)

