The random variable \(X\) has distribution \(\mathrm N(15,a^2)\) and \(\mathrm P(10<X<20)=0.5\). Find the value of \(a\).[2]
The random variable \(Y\) has distribution \(\mathrm B(4,p)\) and \(\mathrm P(Y=1)+\mathrm P(Y=2)=0.5\). Show that \(4p^4-12p^2+8p=1\) and hence find the possible values of \(p\).[4]
On a television quiz show contestants have to select the right answer from one of three alternatives. George decides to do this entirely by guesswork. Use a suitable approximation, which should be stated, to find the probability that George guesses at least \(30\) questions right out of \(100\).[4]
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Answer:(a) \(7.41\) (b) \(0.166\) or \(0.599\) (c) \(0.792\).