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[h]SL 4.8 Binomial distribution, its mean and variance
[q] Binomial Distribution
[a] Binomial distribution refers to situations where we have a certain number of identical (independent) random trials (n), all of which can be viewed as having 2 outcomes, which we may call success & failure, with a fixed probability of success (p). The more basic questions may ask a student to find the prob. of observing ‘x’ successes out of the n trials. Tougher questions may ask about the probability of a range of successes occurring.
[q] Exploration
[a] If you had 7 trials, with a probability of success, P = 0.2 and were asked for P(successes), you may naturally think you must multiply 0.2 by itself 5 times \(0.2^5\), then by 0.8 twice \(×0.8^2\). However, we also must consider all the ways of arranging the 5 successes into 7 places, i.e: (3). Therefore, \(P(x=5)=\frac{7}{5}(0.2)^5(0.8)²≈ 0.00431 \).
We can generalise this result to get the formula: \(P(X=x)=\frac{n}{x}(p)^x(1-p)^{n-x}\).
However, as with PMCC or s.d., we never do this manually.
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