The binomial distribution is given by,
P(X=x) =
![(^{n}C_{x})p^{x} q^{n-x}](https://tex.z-dn.net/?f=%28%5E%7Bn%7DC_%7Bx%7D%29p%5E%7Bx%7D%20q%5E%7Bn-x%7D%20)
q = probability of failure = 1-0.2 = 0.8
n = 100
They have asked to find the probability <span>of obtaining a score less than or equal to 12.
</span>∴ P(X≤12) =
![(^{100}C_{x})(0.2)^{x} (0.8)^{100-x}](https://tex.z-dn.net/?f=%28%5E%7B100%7DC_%7Bx%7D%29%280.2%29%5E%7Bx%7D%20%280.8%29%5E%7B100-x%7D%20)
where, x = 0,1,2,3,4,5,6,7,8,9,10,11,12
∴ P(X≤12) =
![(^{100}C_{0})(0.2)^{0} (0.8)^{100-0} + (^{100}C_{1})(0.2)^{1} (0.8)^{100-1}](https://tex.z-dn.net/?f=%28%5E%7B100%7DC_%7B0%7D%29%280.2%29%5E%7B0%7D%20%280.8%29%5E%7B100-0%7D%20%2B%20%28%5E%7B100%7DC_%7B1%7D%29%280.2%29%5E%7B1%7D%20%280.8%29%5E%7B100-1%7D%20)
+
![(^{100}C_{2})(0.2)^{2} (0.8)^{100-2} + (^{100}C_{3})(0.2)^{3} (0.8)^{100-3}](https://tex.z-dn.net/?f=%28%5E%7B100%7DC_%7B2%7D%29%280.2%29%5E%7B2%7D%20%280.8%29%5E%7B100-2%7D%20%2B%20%28%5E%7B100%7DC_%7B3%7D%29%280.2%29%5E%7B3%7D%20%280.8%29%5E%7B100-3%7D%20)
+
![(^{100}C_{4})(0.2)^{4} (0.8)^{100-4} + (^{100}C_{5})(0.2)^{5} (0.8)^{100-5}](https://tex.z-dn.net/?f=%28%5E%7B100%7DC_%7B4%7D%29%280.2%29%5E%7B4%7D%20%280.8%29%5E%7B100-4%7D%20%2B%20%28%5E%7B100%7DC_%7B5%7D%29%280.2%29%5E%7B5%7D%20%280.8%29%5E%7B100-5%7D)
+
![(^{100}C_{6})(0.2)^{6} (0.8)^{100-6} + (^{100}C_{7})(0.2)^{7} (0.8)^{100-7}](https://tex.z-dn.net/?f=%28%5E%7B100%7DC_%7B6%7D%29%280.2%29%5E%7B6%7D%20%280.8%29%5E%7B100-6%7D%20%2B%20%28%5E%7B100%7DC_%7B7%7D%29%280.2%29%5E%7B7%7D%20%280.8%29%5E%7B100-7%7D)
+
![(^{100}C_{8})(0.2)^{8} (0.8)^{100-8} + (^{100}C_{9})(0.2)^{9} (0.8)^{100-9}](https://tex.z-dn.net/?f=%28%5E%7B100%7DC_%7B8%7D%29%280.2%29%5E%7B8%7D%20%280.8%29%5E%7B100-8%7D%20%2B%20%28%5E%7B100%7DC_%7B9%7D%29%280.2%29%5E%7B9%7D%20%280.8%29%5E%7B100-9%7D)
+
![(^{100}C_{10})(0.2)^{10} (0.8)^{100-10} + (^{100}C_{11})(0.2)^{11} (0.8)^{100-11}](https://tex.z-dn.net/?f=%28%5E%7B100%7DC_%7B10%7D%29%280.2%29%5E%7B10%7D%20%280.8%29%5E%7B100-10%7D%20%2B%20%28%5E%7B100%7DC_%7B11%7D%29%280.2%29%5E%7B11%7D%20%280.8%29%5E%7B100-11%7D)
+
![(^{100}C_{12})(0.2)^{12} (0.8)^{100-12}](https://tex.z-dn.net/?f=%28%5E%7B100%7DC_%7B12%7D%29%280.2%29%5E%7B12%7D%20%280.8%29%5E%7B100-12%7D)
Evaluating each term and adding them you will get,
P(X≤12) = 0.02532833572
This is the required probability.