In a game, Jim throws three darts at a board. This is called a 'turn'. The centre of the board is called the bull's-eye. The random variable $$$X$$$ is the number of darts in a turn that hit the bull's-eye. The probability distribution of $$$X$$$ is given in the following table. It is given that $$$\mathrm{E}(X)=0.55$$$. Jim is practising for a competition and he repeatedly throws three darts at the board. Find the probability that $$$X=1$$$ in at least 3 of 12 randomly chosen turns. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_w21_qp_53 Year：2021 Question No：6(c)

Answer:

$1-\mathrm{P}(0,1,2)=1-\left({ }^{12} \mathrm{C}_{0} 0.3^{0} 0.7^{12}+{ }^{12} \mathrm{C}_{1} 0.3^{1} 0.7^{11}+{ }^{12} \mathrm{C}_{2} 0.3^{2} 0.7^{10}\right)$
$1-(0.01384+0.07118+0.16779)$
$0.747$

Knowledge points:

5.4.2 use formulae for probabilities for the binomial and geometric distributions, and recognise practical situations where these distributions are suitable models (Including the notations B(n,p) and Geo(p). Geo(p) denotes the distribution in which for r = 1,2,3,...)

Answer:

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