A random variable $X$ has a binomial distribution with $n=10$ and probability of success $p$,

Question

A random variable $X$ has a binomial distribution with $n=10$ and probability of success $p$, where $p<0.5$. Given that $\mathrm{P}\left( X=3\,\,\,\mathrm{or}\,\,\,4 \right)=0.2$, write down an equation for the value of $p$, and find this value numerically.[2] It is given that $p=\frac{1}{5}$. The mean and standard deviation of $X$ are denoted by $\mu $ and $\sigma $ respectively. Find $\mathrm{P}\left( \mu -\sigma < X < \mu +\sigma \right)$, correct to $2$ decimal places.[3] Mr Chua attempts an online sudoku puzzle each day. The probability that he manages to solve a puzzle on any given day is $0.75$, independently of any other day. Find the probability that he solves his third puzzle on the eighth day of his attempt.[2] Find the probability that, over a period of $8$ weeks, Mr Chua manages to solve at least $4$ puzzles each week.[2]

Accepted Answer

(a)(i) $0.163$ (ii) $0.77$ (b)(i) $0.00865$ (ii) $0.557$

2022 ASRJC P2 Q9

2022 ASRJC P2 Q9

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