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NSW Y12 Maths Standard 1 NSW Y12 Maths Standard 2 NSW Y12 Maths Advanced NSW Y12 Maths Extension 1 NSW Y12 Maths Extension 2

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NSW Y12 Maths - Extension 1 Binomial Distribution Bernoulli Trials

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Bernoulli Trials Theory

In Bernoulli trials there are only two possible outcomes 'success' and 'failure'. In any stage of an \(n\)-stage binomial trial, the outcomes are \(p\) and \(q=1-p\).\\ In general \(P(x\) successes \()=\left(\begin{array}{l}n \\ x\end{array}\right) p^x q^{n-x}\)\\ \begin{multicols}{2} \textbf{Example 1}\\ A die is rolled ten times: Find the probability that 4 appears on the upper face exactly 3 times \\ \textbf{Example 1 solution}\\ $\begin{aligned} P(3 \text { times }) & =\left(\begin{array}{c} 10 \\ 3 \end{array}\right)\left(\frac{1}{6}\right)^3\left(\frac{5}{6}\right)^7 \\ & =120 \times \frac{5^7}{6^{10}} \\ & =0.155 \end{aligned}$\\ \columnbreak \textbf{Example 2}\\ In a class test there are 8 multiple-choice questions with 5 possible answers, only one of which is correct. How many different ways can you get 6 out of 8 questions correct ?\\ \textbf{Example 2 solution}\\ $\text { From } 8 \text { questions } 6 \text { are correct }\\ \rightarrow\left(\begin{array}{l}8 \\6\end{array}\right)=\dfrac{8 !}{6 ! 2 !}=28$\\ \end{multicols}

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  • Bernoulli Trials - Video - Bernoulli sequence

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