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Recognise repeated Bernoulli trials and evaluate exact probabilities.
For \(X\sim\operatorname{Bin}(n,p)\), \(P(X=r)=\binom nrp^r(1-p)^{n-r}\), \(E(X)=np\), and \(\operatorname{Var}(X)=np(1-p)\).
The trials must have two outcomes, a fixed number of trials, constant success probability and independence. Translate words such as fewer than, at most and at least into integer limits before summing.
Independent trials have constant success probability \(\frac{2}{5}\). Let X count the successes in 7 trials.
\(\binom{7}2p^2(1-p)^{5}=\frac{20412}{78125}\)
Answer: \(\frac{20412}{78125}\).
\(1-P(X=0)-P(X=1)=\frac{65732}{78125}\)
Answer: \(\frac{65732}{78125}\).
\(np=\frac{14}{5}\)
Answer: \(\frac{14}{5}\).
\(np(1-p)=\frac{42}{25}\)
Answer: \(\frac{42}{25}\).
Independent trials have constant success probability \(\frac{1}{2}\). Let X count the successes in 8 trials.
\(\binom{8}2p^2(1-p)^{6}=\frac{7}{64}\)
Answer: \(\frac{7}{64}\).
\(1-P(X=0)-P(X=1)=\frac{247}{256}\)
Answer: \(\frac{247}{256}\).
\(np=4\)
Answer: \(4\).
\(np(1-p)=2\)
Answer: \(2\).
Independent trials have constant success probability \(\frac{1}{5}\). Let X count the successes in 9 trials.
\(\binom{9}2p^2(1-p)^{7}=\frac{589824}{1953125}\)
Answer: \(\frac{589824}{1953125}\).
\(1-P(X=0)-P(X=1)=\frac{1101157}{1953125}\)
Answer: \(\frac{1101157}{1953125}\).
\(np=\frac{9}{5}\)
Answer: \(\frac{9}{5}\).
\(np(1-p)=\frac{36}{25}\)
Answer: \(\frac{36}{25}\).
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Recognise repeated Bernoulli trials and evaluate exact probabilities.
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