To pass, 70% (or MORE) of 20 questions should be answered/guessed correctly.

70% of 20 means that 14 questions or more should be answered/guessed correctly.

In other words, at least 14 questions should be answered/guessed correctly.



The probability to guess correctly for each individual question is  = 0.25.
Guessing provides independent results for each of 2o questions.


So, we have the binomial distribution problem with 20 trials;
the probability of success is 0.25 for each trial.


I will use the MS EXCEL function BINOM.DIST.

Its format is  BINOM.DIST(k, n, p, L).

Here n is the number of trials (20); p is the probability of the individual success (0.25);
k the number of success; L is logical variable, pointing cumulative (if TRUE) or density function (if FALSE).


Using this function, I produced this table below


 k		 BINOM.DIST
------------------------------
 0              0.003171211939
 1		0.021141412926
 2		0.066947807600
 3		0.133895615199
 4		0.189685454866
 5		0.202331151857
 6		0.168609293214
 7		0.112406195476
 8		0.060886689216
 9		0.027060750763
10		0.009922275280
11		0.003006750085
12		0.000751687521
13		0.000154192312
14		0.000025698719    +
15		0.000003426496    +
16		0.000000356927    +
17		0.000000027994    +
18		0.000000001555    +
19		0.000000000055    +
20		0.000000000001    +


The column BINOM.DIST contains the values of probabilies (density function) for the values of successful trial from the first column.

What I need now is the sum of density values for k = 14, 15, 16, 17, 18, 19 and 20.

MS EXCEL allows to get it in couple of clicks.


The ANSWER, i.e. this sum, is about  0.00002951  (rounded).


ANSWER.  The probability to pass this exam is about 0.00002951.