when doing past hsc papers, of the few questions had trouble with were questions such as last year's hsc paper Q7c, and to a lesser extent 2001 hsc paper Q5c (more straight forward than last year's). Does anyone know where I can find additional examples of such questions as these two are the only examples i've ever seen? note that im not talking about regular binomial probability here.
Probability using binomial identities (1 Viewer)
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OldMathsGuy
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Last year's question was brilliant. From memory, the Schaum's book on and combinatorics has a lot of interesting questions.
I remember one that went something like:
6 women and 5 men sit randomly around a table. What is the probability that none of the men will be sitting next to each other?
They are good questions with non-obvious solutions that get you thinking.
Best Regards
OldMathsGuy
I remember one that went something like:
6 women and 5 men sit randomly around a table. What is the probability that none of the men will be sitting next to each other?
They are good questions with non-obvious solutions that get you thinking.
Best Regards
OldMathsGuy
so basically there are no available questions similar to those two past hsc questions? that example u quoted was p&c not binomial. or might there also be some of what im looking for in that book?
thanks
q.
A game is played by n people A1,A2,....An sitting at a table. Each person has a card with their own name on it and all these cards are put in a box. Each person in turn starting with A1 draws a card from random from the box. If the person draws their own card, that person wins and the game ends. Otherwise, the card is returned to the box and the next person draws card. The game continues until someone wins.
Let W be the probability that A1 wins (eventually)
\ Show\ that\ W=p+q^nW\\ \\(2)\ Let\ m\ be\ a\ fixed\ positive\ integer,\ and\ let\ W_{m}\ be\ the\ probability\ that\ A1\ wins\ in\ m\ attempts\ or\ less.\\\\ Given\ that\ \lim_{n\rightarrow \infty}(1-\frac{1}{n})^n=e^{-1}\\\\ Show\ that\ as\ n \to \infty, \frac{W_{m}}{W}\rightarrow 1-e^{-m})
A game is played by n people A1,A2,....An sitting at a table. Each person has a card with their own name on it and all these cards are put in a box. Each person in turn starting with A1 draws a card from random from the box. If the person draws their own card, that person wins and the game ends. Otherwise, the card is returned to the box and the next person draws card. The game continues until someone wins.
Let W be the probability that A1 wins (eventually)
OldMathsGuy
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Yes. You need to find an expression for W [think limiting sum of an infinite series]. Then you need to use that expression for W and show that p + W(q^n) equates to W.
Addition:
You do a very similar thing for part 2) in that you find an expression for Wm [think sum of a geometric series], then you divide that expression by your expression from 1) for W. Simplify and use the limit given to get the final expression of 1 - e^-m
Best Regards
OldMathsGuy
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OldMathsGuy
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Last year's question was as much good application of counting techniques as anything else. The 2001 question was a little more standard (although it sounded from the report like most candidates didn't even attempt the last part). I would suggest the Schaum book on the basis that prods at quite a few ideas - even if most of it is not suitable for Ext 1/Ext 2. Check it out on Google books.
Best Regards
OldMathsGuy