To pass, 60% (or MORE) of 10 questions should be answered/guessed correctly.

60% of 10 means that 6 questions or more should be answered/guessed correctly.

In other words, at least 6 questions should be answered/guessed correctly.



The probability to guess correctly for each individual question is 1/4 = 0.25.
Guessing provides independent results for each of 10 questions.


So, we have a binomial distribution problem with 10 trials;
the probability of success is 0.25 for each individual trial.


We want to find the probability of having 6 or more successes.


It can be calculated in several different ways.



(1)  First, you may use spreadsheet like MS excel or any calculator and compute the probability of success
     of binomial distribution for values of trials k = 6, 7, 8, 9 10, and add them

          P = P(6) + P(7) + P(8) + P(9) + P(10).    (1)

     Each addend P(k) is  P(k) = .


     Usually this way is considered as time consuming and not economical.


     For more fast calculations, usually the complementary probability formula is used

        P = 1 - (P(0) +P(1) + P(2) + P(3) + P(4) + P(5)).   (2)


     This sum in parentheses from 0 to 5 is called "cumulative sum".

     For cumulative sums, there are special calculator functions that provide fast economical calculations.



(2)  Following formula (2), you can use the standard Excel function BINOM.DIST in the mode, 
     which provides the cumulative sum as the output.

     About this function read from this web-site
     https://corporatefinanceinstitute.com/resources/excel/binomial-distribution-excel/

     Then the formula to get the answer in this problem is  P = 1 - BINOM.DIST(5, 10, 0.25, TRUE)



(3)  Alternatively, you may use a regular calculator TI-83/84 and its standard function
     binomcdf, which produces cumulative function output, too.

     About this function read from this web-site
     https://www.mathbootcamps.com/binomial-probabilities-ti-83-or-84-calculator/

     Then the formula to get the answer in this problem is  P = 1 - binomcdf(10, 0.25, 5).



(4)  Finally, you can use an online calculator (free of charge) at this web-site
     https://stattrek.com/online-calculator/binomial.aspx

     This calculator has simple and convenient interface, so any student, even 
     a beginner, can easily work with it.



    The  ANSWER to the problem's question is  P = 0.01973  (rounded).