Each question has #4# possible answers only one of which is correct, if I do not have the knowledge or aware about the answer and we depend upon guess
, the probability of getting that particular answer right is #1/4#.
However, there are #22# question each of which has a probability of being correctly answered is #1/4#, the probability of exactly #r# of them being answered correctly #P(r)# is given by
#P(r)=color(while)(x)^nC_rp^rq^((n-r))#, where #q=1-p#
Further, as we desire the guessing of at most #3# answers correctly i.e. #P(r<=3)#, this is
#P(r<=3)=P(0)+P(1)+P(2)+P(3)#
= #color(white)(x)^22C_0(1/4)^0(3/4)^22+color(white)(x)^22C_0(1/4)^1(3/4)^21+color(white)(x)^22C_0(1/4)^2(3/4)^20+color(white)(x)^22C_0(1/4)^3(3/4)^19#
= #(3/4)^22+22xx1/4xx(3/4)^21+(22xx21)/(1xx2)xx(1/4)^2xx(3/4)^20+(22xx21xx20)/(1xx2xx3)xx(1/4)^3xx(3/4)^19#
= #0.00178381+0.01308125+0.04578437+0.10174305#
= #0.16239248#
Hence, the probability that you will guess at most #3# answers correctly is #0.1624#
