Cumulative Probabilities and Calculator Use
Cumulative Probabilities and Calculator Use
Binomial cumulative distribution function
For \(X\sim\mathrm{B}(n,p)\) and an integer \(x\) with \(0\le x\le n\),
\[P(X\le x)=P(0\le X\le x)=\sum\limits_{k=0}^{x}\binom nkp^k(1-p)^{n-k}\]
A binomial cumulative command returns \(P(X\le x)\). Express the required probability in this form before using the calculator. In the table, \(a,b\) are integers in \([0,n]\); set \(P(X\le-1)=0\).
| Required probability | Cumulative form | Condition |
|---|---|---|
| \(P(X\le n)\) | \(1\) | All binomial models |
| \(P(X<a)\) | \(P(X\le a-1)\) | Integer boundary |
| \(P(X>a)\) | \(1-P(X\le a)\) | Integer boundary |
| \(P(X\ge a)\) | \(1-P(X\le a-1)\) | Integer boundary |
| \(P(a\le X\le b)\) | \(P(X\le b)-P(X\le a-1)\) | \(a\le b\) |
| \(P(a<X<b)\) | \(P(X\le b-1)-P(X\le a)\) | \(a<b\) |
Translate the event into an integer range before using technology. For integer-valued \(X\) and integer \(k\), \(P(X<k)=P(X\le k-1)\) and \(P(X\ge k)=1-P(X\le k-1)\). A non-integer threshold must first be converted to the actual integer values permitted by the inequality.
For a least-number-of-trials question, first justify how the probability changes with the number of trials. With a fixed success probability and a fixed target number of successes, the probability of reaching at least that target cannot decrease when an extra trial is added. In this monotonic case, check that the proposed integer satisfies the requirement and the preceding integer fails. If monotonicity is not established, checking only the preceding integer does not prove a global minimum; consider all admissible smaller values or give another justification. Record the model and event, not only a calculator screen.
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