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NSW Y12 Maths - Extension 1 Binomial Distribution Binomial Distribution

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Binomial Distribution Theory

\(X \sim B(n, p)\) indicates a random variable \(X\) that has a binomial distribution with \(n\) trials and a probability of success of \(p\).\\ \textbf{Example 1}\\ Giver a coin is tossed 4 times then \(X \sim B\left(4, \frac{1}{2}\right)\). what is the probability of obtaining 2 heads and 2 tails?\\ \textbf{Example 1 solution}\\ $\begin{aligned} (p+q)^4&=\left(\begin{array}{l} 4 \\ 0 \end{array}\right) p^4+\left(\begin{array}{l} 4 \\ 1 \end{array}\right) p^3 q+\left(\begin{array}{l} 4 \\ 2 \end{array}\right) p^2 q^2+\left(\begin{array}{l} 4 \\ 3 \end{array}\right) pq^3+\left(\begin{array}{l} 4 \\ 4 \end{array}\right) q^4 \\ p(2 \text { Heads })&=\left(\begin{array}{l} 4 \\ 2 \end{array}\right)\left(\frac{1}{2}\right)^2\left(\frac{1}{2}\right)^2 \\ &=6 \times \frac{1}{4} \times \frac{1}{4} \\ &=\frac{3}{8} \end{aligned}$\\ \textbf{Example 2}\\ A die is tossed 5 times. What is the probability of obtaining 3 sixes? \\ \textbf{Example 2 solution}\\ $\begin{aligned} X &\sim B\left(5, \frac{1}{6}\right) \\ P(3 \text { sixes })&=\left(\begin{array}{l} 5 \\ 3 \end{array}\right)\left(\frac{1}{6}\right)^3\left(\frac{5}{6}\right)^3 \\ &=10 \times \frac{1}{216} \times \frac{125}{216} \\ &=0.027 \end{aligned}$\\

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Videos

Videos relating to Binomial Distribution.

  • Binomial Distribution - Video - Binomial distribution introduction

  • Binomial Distribution - Video - The binomial Probability Formula

  • Binomial Distribution - Video - Binomial Probability Distribution formula Worked Example

  • Binomial Distribution - Video - Binomial Probability formula at most and at least

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NSW Year 12 Maths Extension 1 - Trig Equations

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NSW Year 12 Maths Extension 1 - Calculus

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NSW Year 12 Maths Extension 1 -Differential Equations

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NSW Year 12 Maths Extension 1 -Projectile Motion

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NSW Year 12 Maths Extension 1 -Binomial Distribution

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