<PRE><B>Question 1206327</B>
.
A quiz consists of 20 multiple-choice questions, each with 4 possible answers. 
For someone who makes random guesses for all of the answers, 
find the probability of passing if the minimum passing grade is 70 %.
~~~~~~~~~~~~~~~~~~~~~


&lt;pre&gt;
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 {{{1/4}}} = 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 &lt;U&gt;ANSWER&lt;/U&gt;, i.e. this sum, is about  0.00002951  (rounded).


&lt;U&gt;ANSWER&lt;/U&gt;.  The probability to pass this exam is about 0.00002951.
&lt;/pre&gt;

Solved.


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To learn more about function BINOM.DIST, look into this web-page


https://support.microsoft.com/en-us/office/binom-dist-function-c5ae37b6-f39c-4be2-94c2-509a1480770c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.



</PRE>