In a binomial probability distribution, if the probability of success for an event is #p# (i.e. for failure the probabilty is #(1-p)#),
the probability of getting #r# successes out of a total trial of #n# events i.e. #P(n,r)# is given by
#P(n,r)=(n!)/(r!(n-r)!)p^r(1-p)^(n-r)#
In the given case we have #p=0.4#, #n=13# and at least #7# success means success of #7# or more.
Hence desired probaility #P(r>=7)#
= #P(7,13)+P(8,13)+P(9,13)+P(10,13)+P(11,13)+P(12,13)+P(13,13)#
= #(13!)/(7!6!)xx0.4^7xx0.7^6+(13!)/(8!5!)xx0.4^8xx0.7^5+(13!)/(9!4!)xx0.4^9xx0.7^4+(13!)/(10!3!)xx0.4^10xx0.7^3+(13!)/(11!2!)xx0.4^11xx0.7^2+(13!)/(12!1!)xx0.4^12xx0.7^1+(13!)/(13!0!)xx0.4^13xx0.7^0#
= #0.13117+0.06559+0.02429+0.00648+0.00118+0.00013+0.00001#
= #0.22885#
