The random variable $X$ follows a binomial distribution $\mathrm{B}\left( 10,p \right)$. Given t

Question

The random variable $X$ follows a binomial distribution $\mathrm{B}\left( 10,p \right)$. Given that $X$ has two modes, $X=4$ and $X=5$, find the exact value of $p$.[2] Given instead that $\mathrm{P}\left( X\le 9 \right)=\frac{1023}{1024}$, find the exact value of $p$.[2] The random variable $Y$ follows a binomial distribution $\mathrm{B}\left( 500,0.5 \right)$ A sample of $30$ independent values of $Y$is recorded. Find the probability that all the values recorded are less than or equal to $256$.[2] The mean of the $30$ values is calculated. Estimate the probability that this sample mean is less than or equal to $256$, stating clearly the approximation used.[3] Explain why the probability found in part (ii) is larger than that found in part (i).[1]

Accepted Answer

(a)(i) $p=\dfrac5{11}$ (ii) $p=\dfrac12$ (b)(i) $0.0000514$ (ii) $0.998$ (iii) The event in part (i) is a subset of the event in part (ii)

2017 CJC P2 Q7

2017 CJC P2 Q7

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