MATH1011 Chap.2 Counting Methods and Basic Probability
Counting Methods and Basic Probability
Deciding what a rearrangement would change
A counting question is usually won or lost in its first ten seconds, well before any arithmetic happens. Read the stem and settle two things about the objects being picked. Would writing the same chosen objects in a different order count as a different result? Seating people along a bench, awarding first and second place, and building a passcode all answer yes.
Naming a committee, dealing a hand, or writing down a subset all answer no. Then ask whether one object may be picked more than once: rolling a die repeatedly allows it, drawing names from a hat and keeping them does not.
Getting this wrong is not a small error.
Taking three items from eight in order, with nothing repeated, gives 336 results, while taking three from eight as a plain handful gives 56. The two answers stand a factor of six apart, so misreading one word of the stem changes the answer completely rather than slightly.
Rebuilding the selection formulas from scratch
You never have to remember four formulas if you can reconstruct two of them.
Picking three of eight items in order gives eight ways to fill the first place, seven for the second because one item has left circulation, and six for the third. Those numbers multiply because the three choices are consecutive steps inside one act of picking, and 8 times 7 times 6 is 336. Now rub out the labels first, second and third.
Each group of three has already been written down six separate times, once per arrangement of those same three items, so dividing 336 by 6 leaves 56 genuinely different groups.
Two savings follow at once. Choosing which three of eight to take is the same act as choosing which five to leave behind, so both counts come out as 56 and you may always work with whichever of the two numbers is smaller.
And counts join in exactly two ways. Pieces that follow one another inside a single selection are multiplied. Pieces that compete with one another, because at most one of them can describe the finished result, are added instead.
Listing a sample space that favours nothing
Probability enters as soon as the counting is trustworthy.
An experiment, in the narrow meaning used here, is one run of something whose individual result nobody can call ahead of time, even though the menu of possible results was settled before the run began. Collect that menu into a set and you have the sample space; any happening you care about is one of its subsets. Everything then hangs on how finely the menu is written.
Toss two coins and you might record three results: two heads, one of each, two tails. Those three are not interchangeable, since one of each arises in two ways and the other two in one way apiece.
Record the coins separately and you get four results that genuinely are interchangeable, and only a list like that may be divided into.
Probability as division, and the shortcut through the opposite
With an even list in hand, a probability is a fraction of two counts, favourable on top and total underneath.
Both counts must be written in the same language: if the whole space was counted as unordered handfuls, the favourable ones may not be counted as ordered sequences. Take a bag holding 3 red and 5 green counters and draw two together. There are 28 pairs altogether.
Both red accounts for 3 of them and both green for 10, so a matching pair has probability 13 over 28.
The remaining 15 pairs are the mixed ones, and you can reach that number independently: 3 choices of red times 5 choices of green is 15, and 13 plus 15 returns 28. Chasing an agreement like that is worth the thirty seconds on any question, because it is free evidence that both counts were taken the same way.
It also demonstrates the standard shortcut. Whenever an event is described by the words at least one, its opposite is a single tidy case rather than several, so count the opposite, divide, and subtract the result from 1.
What this chapter covers
- 01
Order or no order, repetition or none
- 02
Permutations and the falling product behind them
- 03
Combinations as ordered counts with the orders divided out
- 04
Sample spaces, events and equally likely listing
- 05
Probability as one count over another
- 06
Counting the opposite event instead
Worked example · free
Three keys from a drawer, counted twice and then turned into chance
- 2Count the ordered sequences by filling three places in turn, remembering that a key taken is not put back.
- 2Convert that ordered total into a count of selections by dividing out the arrangements of each group.
- 3Count the selections holding exactly one working key by choosing the working key and the two useless ones separately, then divide.
- 2Handle at least one through its opposite, and state both probabilities as fractions in lowest terms.
Key terms
- Permutation
- A permutation is an arrangement in which order carries information, so exchanging two of the chosen objects yields a different arrangement. Taking r objects in order from n distinct ones gives a falling product of exactly r factors, starting at n and dropping by one each time.
- Sample Space
- The sample space is the set of every result one run of an experiment can produce, written so that exactly one of them occurs on each run. Its size supplies the denominator of any probability worked out by counting.
- Equally Likely Outcomes
- Equally likely outcomes are results listed at a fineness that leaves none of them favoured over any other. Only a sample space written this way may have probabilities read off it by dividing one count by another.
- Complementary Event
- A complementary event consists of precisely the outcomes its partner leaves out, so the two probabilities add to 1. Counting it is the standard route through any event worded as at least one.
Counting Methods and Basic Probability FAQ
How do I tell a permutation question from a combination question?
Picture the result already chosen, swap two of the chosen items, and ask whether anything has changed. If the swap produces a different outcome, such as a different seating plan or a different passcode, you are counting permutations. If the same people or cards have merely been written down in another order, it is a combination and the orderings have to be divided out.
Why do some answers start by counting what I do not want?
Because the phrase at least one covers a pile of separate cases while its opposite, none at all, is a single case. Counting that one case and taking its probability away from 1 replaces three or four counts with one, and the same manoeuvre rescues questions worded at most, which flip into at least.
Exam move
Practise the decision on wording alone before touching arithmetic. Take twenty question-bank stems and, for each one, write only two letters: O or U for whether order counts, and R or N for whether repetition is allowed.
Mark them against the published answers by comparing the shape of the formula rather than the number it produces, then rewrite the five you labelled wrongly, copying out beside each the exact phrase in the stem that should have decided it.
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