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Using Binomial Probability Tables
Discrete Mathematics · Chapter 8: Binomial Probability, Recurrence Relations, and Generating FunctionsSection 8.4 · lesson 49 of 87video coming soon
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In this lesson
Lookup and arithmetic instead of the formula - and where tables run out.
- Read a cumulative binomial table
- Use symmetry for p above 0.50
- Get exact, at-least and interval probabilities by arithmetic
- Bracket untabulated values, and know the limits
Transcript
Everything said in the video, section by section (1,371 words). Read along, or search it.
Binomial Probability, Recurrence Relations, and Generating Functions
Using Binomial Probability Tables
This section is about binomial probability tables. We'll read them, combine their entries with simple arithmetic to answer questions they don't tabulate outright, and see exactly where the table method runs out. No formula computation happens here. Every example is solved by lookup, and arithmetic on table values.
The Opening Problem
The Same Lookups, Lot After Lot
A supplier-quality engineer at an electronics manufacturer evaluates each incoming lot of components under an acceptance-sampling plan, modeled on a published inspection standard. The plan samples twenty units from the lot. She needs to know how likely it is that two or fewer are defective, if the lot's true defect rate is five percent. She could compute that from the binomial formula. But the whole point of a standard is that the same few combinations of n and p, get used lot after lot. So rather than recompute the same probability every time, quality engineers, exam candidates without a calculator, and standards readers reach for a precomputed table.
Table 8.4.1
Cumulative Probabilities P(X ≤ k), n = 20
A standard binomial table is organized in blocks, one for each tabulated value of n. Within a block, columns are indexed by p, typically values like zero point zero one, zero point zero five, zero point one, and on up to one half. Rows are indexed by k, from zero up to n. Most tables record the cumulative probability, the chance that X is at most k, rather than the chance of exactly k. A cumulative table can be combined to answer more kinds of questions. This is part of the block for n equal to twenty. Every value shown was recomputed, and matches to four decimal places.
Worked Example 8.4.1a
A Direct Lookup
Now the engineer's question. What's the probability that X is at most two, for n equal to twenty and p equal to zero point zero five? Find the block for n equal to twenty. Move across to the column for p equal to zero point zero five. Then move down to the row for k equal to two. The entry at the intersection is zero point nine two four five. So there's about a ninety-two percent chance of finding two or fewer defective units. No arithmetic was needed beyond locating a row and a column.
Theorem 8.4.1
Symmetry: Why Tables Stop at 0.50
Published tables usually list only p up to one half. That's because of a symmetry. If X is binomial with parameters n and p, then n minus X is binomial with parameters n and one minus p. The reason is simple. Relabel success as failure, and failure as success, on every trial. What counted as a success under p is now a failure under one minus p. So the number of relabeled successes is exactly n minus X. And relabeling changes neither the fixed number of trials, nor their independence. The event X at most k is then the same as n minus X at least n minus k. That turns into a cumulative lookup through the complement rule.
Worked Example 8.4.1b
A p Above 0.50, Read by Symmetry
Here's the symmetry in use. A batch test passes each of twenty units independently, with probability zero point nine. What's the chance that at most seventeen pass? Zero point nine isn't a column. So let Y, twenty minus X, count the units that fail. Y is binomial with n twenty and p zero point one, which is tabulated. At most seventeen passing is the same as at least three failing. That's one minus the probability that Y is at most two. The table gives zero point six seven six nine. So the answer is zero point three two three one.
8.4.2 Lookup Plus Arithmetic
Every Query From Cumulative Entries
A cumulative table only lists one kind of quantity, the chance of at most k. But most useful questions take other forms. The chance of exactly k is the cumulative entry at k, minus the entry at k minus one. The chance of at least k is one minus the entry at k minus one. And the chance of landing between a and b, inclusive, is the entry at b, minus the entry at a minus one. No exponent, factorial, or binomial coefficient is ever evaluated by hand. The table has already absorbed that computation.
Worked Examples 8.4.2a to 8.4.2c
3 Queries, 3 Kinds of Arithmetic
Three quick examples, all from the n equals twenty block. Exactly three successes, with p zero point one, is zero point eight six seven, minus zero point six seven six nine. That's zero point one nine zero one. At least four, with p zero point zero five, is one minus zero point nine eight four one. That's zero point zero one five nine. And between two and five, with p zero point one, is zero point nine eight eight seven, minus zero point three nine one seven. That's zero point five nine seven. The pattern is always the same. Express the event with one or two cumulative probabilities, look each up, and combine them.
8.4.3 A Missing p
Bracket It - Don't Just Interpolate
No table can list every possible combination of n and p. When the situation falls off the grid, the method can stop giving an exact answer. Suppose p is zero point zero seven, between two columns. Linear interpolation between them gives only an approximation, of unknown precision. But there's a rigorous alternative. For fixed n and k, the cumulative probability strictly decreases as p increases. Here's why. Drive both experiments with the same uniform random numbers, and call trial i a success whenever its number is below p. Then every success under the smaller p is also a success under the larger one. So the count under the larger p is always at least as big, and at most k becomes less likely.
Worked Example 8.4.3
An Exact Bracket for p = 0.12
That gives an exact bracket. Suppose we need the chance that X is at most four, for n equal to twenty and p equal to zero point one two, which isn't tabulated. Zero point one two lies between zero point one and zero point one five. So by monotonicity, the answer lies between those two columns' entries. The table gives zero point eight two nine eight, and zero point nine five six eight. So the true value is strictly between them. That's exactly justified, a genuine advantage over naive interpolation. Though it gives a range, not a single number.
2 Harder Limits
A Missing n, and Rounding
Two harder limits remain. A missing n has no comparable fallback. n is a whole number of trials, fixed by the experiment's design. And blocks for different n describe entirely different sample spaces. If a problem needs n equal to twenty-seven, and no such block exists, the table method has reached its limit. Use a bigger table, the formula, or software. And rounding can compound. Tables round each entry, here to four decimals. Subtracting two nearly equal entries amplifies that. For example, with p of zero point one five, the chance of exactly nine is zero point nine nine nine eight, minus zero point nine nine eight seven. That's zero point zero zero one one. But each entry could be off by up to half a unit in the last place. So the difference is uncertain by about zero point zero zero zero one, close to a tenth of the answer itself.
Table 8.4.2
Table, Formula or Software?
This table compares the three ways to get a binomial probability. A printed table handles only tabulated pairs, or brackets between them. It needs no device, and it's fast when the pair is listed. It's typical in calculator-free exams and published standards. The formula handles any n and p, but needs a calculator and several steps. And software, or a spreadsheet function, handles any case in a single call. It's routine practice today. So the table's core limitation is coverage, not speed.
Takeaways
Look Up, Combine, Know the Limits
To summarize. A binomial table has one block per n, with columns for p and rows for k, and usually records cumulative probabilities. The symmetry property lets a table that stops at one half answer questions for larger p. Exact, at least, and interval questions all come from cumulative entries, by subtraction, complements, or differences. And a missing p can be bracketed exactly, using monotonicity. A missing n can't. Here's something to think about. Is that asymmetry really about p being continuous and n being discrete? After all, k is a discrete integer too, and a cumulative table brackets between neighboring k values perfectly well. What property does the monotonicity argument actually depend on? And does that property have any analogue when n is the parameter that varies?
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About Using Binomial Probability Tables
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