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I made a program to guess on a multiple choice test with an unlimited amount of questions. I want to know the chances of getting 5 questions correct in a row.

There are 4 choices on each question. The program guesses once every question. The test is not passed until the computer guesses 5 questions correct in a row.

PS, if you were going to consider an amount less than infinite, use 100 or 1000 please. If you are here to help can you also explain instead of just answering? I'm here to learn from this, not to just take your answers...

Nick Cox
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Crayzie
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  • As $n\to\infty$ we have $P(5+ \text{ correct in a row at least once})$ using random guessing goes to $1$. – Glen_b Feb 13 '15 at 00:43
  • So about 1/1000? – Crayzie Feb 13 '15 at 01:10
  • Sorry, I have no idea what you're asking there. Simulation puts the probability in $n=100$ case (call that $p_{100}$ say) at about 7% and the $n=1000$ case at about 52% ($p_{1000}\approx 0.52$). A useful lower bound on some other $n>100$ would be $1-(1-p_{100})^{n/100}$ (using $p_{100}=0.07$ suggests $p_{1000}>0.516$, but the uncertainty in the estimate of $p_{100}$ would impact such estimates). One can construct a similar lower bound estimate based off $\hat{p}_{1000}$ which should be fairly accurate for large $n$. – Glen_b Feb 13 '15 at 01:17
  • I ran tests, and that is about right. I looped 100000 true/false questions and got 50071 correct answers, out of that I got 1601 five in row's. – Crayzie Feb 13 '15 at 01:32
  • Why is the `survey` tag there? To answer this question algebraically, I'd probably look at generating functions or trying to get numerical solutions to the recurrence relations. – Glen_b Feb 13 '15 at 01:32
  • Come to think of it, for $n<1000$ one can use the same formula to get a useful upper bound running back from $n=1000$. If one does that for $n$ between 100 and 1000, the answers from the (somewhat fuzzy due to simulation uncertainty) upper and lower bounds are almost identical. – Glen_b Feb 13 '15 at 01:48
  • Do you know what a Markov Chain is? Do you know what a transition matrix is? I think I can write an answer using these ideas but it won't help if you don't know them. – Yair Daon Feb 17 '15 at 03:57

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