Question

g, Assume that random guesses are made for 15 multiple chioce questions on an SAT test and that there are 5 choices on each question with the probability of success 0.20. Find the probability that the number of correct answers is at most 6 (This problem meets all the requirements of a binomial situation.)

Answer 1

Answer: 0.9819

Step-by-step explanation:

Binomial probability formula :-

$P(X)=^nC_x \ p^x\ (1-p)^{n-x}$, where P(x) is the probability of getting success in x trials, n is total number of trials and p is the probability of getting succes in each trial.

Given : The probability of success :  $p=0.20$

The total question answered : n= 15

Now, the probability that the number of correct answers is at most 6 is given by :-

$P(x\leq6)=P(0)+P(1)+P(2)+P(3)+P(4)+P(5)+P(6)\\\\= ^{15}C_{0}(0.20)^0(0.8)^{15}+^{15}C_{1}(0.20)^1(0.8)^{14}+^{15}C_{2} (0.20)^2(0.8)^{13}+^{15}C_{3} (0.20)^3 (0.8)^{12}+^{15}C_{4} (0.20)^4(0.8)^{11}+^{15}C_{5} \ (0.20)^5\ (0.8)^{10}+^{15}C_{6} \ (0.20)^6\ (0.8)^{9}\\\\=(0.8)^{15}+(15)(0.2)(0.8)^{14}+105(0.2)^{2}(0.8)^{13}+455(0.2)^{3}(0.8)^{12}+1365(0.2)^{4}(0.8)^{11}+3003(0.2)^{5}(0.8)^{10}+5005(0.2)^{6}(0.8)^{9}=0.981941193015\approx0.9819$

Hence, the probability that the number of correct answers is at most 6 = 0.9819