i. All the possible answers in Quiz 1 by chance can be represented by the terms in the expansion of (0.25+0.75)^8, where 0.25 is the probability of a correct choice of answer for each individual question and 0.75 is the probability of an incorrect choice of answer. The binomial expression has a value of 1, which is a probability of 1, i.e., certainty, made up of the sum of all the terms. Let p=0.25 and 1-p is 0.75. The expansion is p^8+... + 8Cr*p^r(1-p)^(8-r), where r indicates the rth term between r=0 and 8 and 8Cr is the combination function nCr for n=8. The value of 8Cr is given by (8*7*...*(8-r+1))/(1*2*...*r). The coefficients are, in order: 1, 8, 28, 56, 70, 56, 28, 8, 1, the 8th row of Pascal's triangle. The binomial terms start with the probability of 8 correct answers, then 7 correct 1 incorrect, 6 correct 2 incorrect, 5 correct 3 incorrect, and so on.
The probability of answering incorrectly 3 question is (3/4)^3=27/64, so the probability of answering at least one question in 3 correctly by chance is 1-27/64=37/64, in other words, this is the probability that at most 3 questions are attempted before obtaining a correct answer. This is better than an evens chance (50%), so is more likely than unlikely.
ii. In Quiz 2 p=1/5 or 0.20 and n=15. The probability of less than 4 correct answers is the sum of the last four terms of the binomial series for 15, which give all incorrect, 1, 2, 3 correct. The coefficients are respectively 1, 15, 105, 455. The p terms alone are 0.03518, 008796, 0.002199 and 0.0005498. When we multiply by the coefficients and add the terms together we get 0.6482 or 64.82% probability.
iii. The probability of 6 or more correct answers in Quiz 1 is 1-(probability of up to 2 incorrect answers)=1-(0.75^8+8*0.75^7*0.25+28*0.75^6*0.25^2)=1-0.6785=0.3215 or 32.15%. This is less than 1/3, so the odds are 2:1 against Siti getting 6 or more correct answers by chance alone, which makes it reasonably unlikely.
iv and v. The problem says "before the 8th question", so there need to be 5 correct answers within the first 7 questions. Using p=0.2, the probability of getting 5 correct answers out of 5 is 0.2^5=0.00032; the probability of getting 5 correct out of 6 is 6*0.2^5*0.8=0.001536; and the probability of 5 out of 7 is 21*0.2^5*0.8^2=0.0043008. Therefore the probability of getting 5 correct before the 8th question is the sum of these=0.0061568 or 0.62%, a most unlikely occurrence, less than 1 in 162.
