Have you ever attended a test without studying anything whatsoever before it? Well your boy Cris is here to tell you something: you most certainly failed. Or, at least, you would have most certainly failed in a world fully governed by A-level Mathematics. I managed to find last year's STEP today, and I decided to take a look at some of the problems. Apart from the tedious, VERY TEDIOUS calculations, I found a nice problem that many people will emphasize with. This problem was the last Probability exercise in Step 1 last year.
A multiple-choice test consists of five questions. For each question, n answers are given
(n > 2) only one of which is correct and candidates either attempt the question by choosing
one of the n given answers or do not attempt it.
For each question attempted, candidates receive two marks for the correct answer and lose
one mark for an incorrect answer. No marks are gained or lost for questions that are not
attempted. The pass mark is five.
Candidates A, B and C don’t understand any of the questions so, for any question which
they attempt, they each choose one of the n given answers at random, independently of their
choices for any other question.
(i) Candidate A chooses in advance to attempt exactly k of the five questions, where
k = 0, 1, 2, 3, 4 or 5. Show that, in order to have the greatest probability of passing
the test, she should choose k = 4 .
(ii) Candidate B chooses at random the number of questions he will attempt, the six
possibilities being equally likely. Given that Candidate B passed the test find, in
terms of n, the probability that he attempted exactly four questions.
(iii) For each of the five questions Candidate C decides whether to attempt the question
by tossing a biased coin. The coin has a probability of \(\frac{n}{n+1}\) of showing a head, and
she attempts the question if it shows a head. Find the probability, in terms of n, that
Candidate C passes the test.
Solution:
(i) The probability of getting a correct answer to a question is \(\frac{1}{n}\). If k is smaller than 3, then the passing score isn't achieved in any scenario, so the probability of passing for k<3 is 0.
If k=3, then the probability is \(\frac{1}{n^3}\).
If k=4, then the probability of passing is \({4}\choose{3}\)\(\frac{n-1}{n^4}+\frac{1}{n^4}=\frac{4n-3}{n^4}\).
If k=5, then the probability of passing is \( {5}\choose{4}\)\(\frac{n-1}{n^5}+\frac{1}{n^5}\). In the last case, the minimum number of possible correct answers is 4, because otherwise the passing mark isn't achieved. Let us now compare the probabilities we found.
$$\frac{4n-3}{n^4}>\frac{1}{n^3}⟺4n-3>n⟺n>1$$.
$$\frac{4n-3}{n^4}>\frac{5n-4}{n^4}⟺4n^2-3n>5n-4⟺4(n-1)^2>0$$
Thus, the biggest probability of passing is when k=4.
(ii) We shall use Bayes' theorem for this one. Bayes' theorem states that, if we want to calculate the probability that the even B happens given that another event A happened before, then the probability required is \(P(B|A)=\frac{P(B⋂A)}{P(A)}\). Thus, the probability of B passing is $$P(k=4|pass)=\frac{P((k=4)⋂pass)}{P(pass)}=$$
$$=\frac {\frac{4n-3}{n^4}}{\frac{5n^2+2n-4}{n^5}}=$$
$$=\frac{4n^2-3n}{5n^2+2n-4}$$
(iii) \(P(C|pass)=\frac{10n^3}{(n+1)^5}*\frac{1}{n^3}+\frac{5n^4}{(n+1)^5}*\frac{4n-3}{n^4}+\frac{n^5*(5n-4)}{n^{5}*(n+1)^{5}}=\frac{25n-9}{(n+1)^5}\).
The problem itself was rather easy, and the calculations were simple enough to be done quick enough. The Step this year had some really interesting geometry problems, and, from what I've heard, had a monster algebra problem. While I find Step questions insanely interesting, I find the calculations required a bit tedious.
I attended a contest similar to the one described in the problem when I was younger. I am pretty sure many kids across the world, too, participated at the Mathematical Kangaroo competition. It was a rather good mental exercise for kids, in which you were asked to solve 30 questions in an hour and a half or something. I never did good in multiple choice tests, so I didn't have insanely good results at it; if I recall correctly, I once started crying desperately in one such contest, and the teacher assisting us was afraid to take my paper when the time concluded. Yikes!
Cris.
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