Math: Data, Chance and Counting

This is Unit Fifteen, part two, of ACT Preparation, and the last of the mathematics. Part one closed by predicting from a line inside the data; this part turns to chance and counting. The habit carries over: ask what the numbers describe. For a probability, that means asking which group the question limits you to. By the end you will be able to find probabilities with and, or, and the overlap. To find the chance of at least one. To read a conditional probability from a table. To count with permutations and combinations. And to find a term of a sequence.

Eight entries on probability: the basic ratio and the rules built on it.

Favorable over total.

ACT 1421 Probability Basics

When outcomes are equally likely, the probability of an event is the number of favorable outcomes over the total. Probabilities run from zero to one, and the probability of not A is one minus the probability of A. Rolling a die, at least five has probability one third, so less than five has two thirds. Count the whole sample space first.

Independent events.

ACT 1422 Independent Events

Events are independent when one does not change the probability of the other, and then the probability of both is the product. Two heads in two flips is one quarter; a six and a head is one twelfth. Drawing without replacement breaks independence.

Events that cannot happen together.

ACT 1423 Mutually Exclusive Events and the Addition Rule

Mutually exclusive events cannot happen together, so the probability of either is the sum. In general, add and subtract the overlap: a king or a queen is eight fifty-seconds; a king or a heart is four plus thirteen minus one, sixteen fifty-seconds. Mutually exclusive and independent are different ideas.

The chance of at least one, step by step.

ACT 1424 The Chance of At Least One

Name the opposite event, no successes; find its probability by multiplying the chance of failure across independent trials; and subtract from one. A shooter who makes seventy percent misses all three with probability 0.3 cubed, 0.027, so makes at least one with 0.973. At least one six in four rolls is one minus five sixths to the fourth, about 0.52. Adding the chances gives 2.1, above one, which signals the error: the events overlap.

Or, with an overlap, step by step.

ACT 1425 Or, With an Overlap

Decide whether the events can happen together; if not, add; if so, add and subtract the probability of both. In a club of forty, twenty-two play soccer, fifteen tennis and seven both, so soccer or tennis is thirty fortieths, three quarters, and ten play neither. An even number or a five on a die has no overlap: four sixths. Without the subtraction, the seven who play both are counted twice.

Conditional probability.

ACT 1426 Conditional Probability and Two-Way Tables

A conditional probability, A given B, restricts attention to the cases where B is true: the probability of both, over the probability of B. Of fifty seniors, thirty drive, so driving given senior is three fifths. A given B and B given A usually differ, because their denominators are different groups.

Reading a table, step by step.

ACT 1427 Conditional Probability From a Table

Find the given condition, after a word such as given, if or among; use its row or column total as the denominator; and divide the count that also meets the other condition. In an invented survey of 200, fifty-four of ninety juniors bring lunch, and ninety-eight bring lunch in all. Lunch given junior is 0.6; junior given lunch is fifty-four ninety-eighths, about 0.55; junior and lunch is 0.27. The three share a numerator and differ only in the denominator.

Expected value.

ACT 1428 Expected Value

The expected value is the long-run average outcome: multiply each outcome by its probability and add. A game paying $10 with probability 0.2 has an expected value of $2 a play. The expected value need not be a possible outcome, and a cost to play subtracts from it.

Six entries on counting and sequences, and the whole unit in one data set.

Counting choices.

ACT 1429 The Multiplication Principle

If one choice can be made in m ways and a second in n ways, the two together can be made in m times n ways. Four shirts and three pairs of pants make twelve outfits, and two pairs of shoes make twenty-four. Choices made together multiply; choices made instead of each other add.

Permutations and combinations.

ACT 1430 Permutations and Combinations

A permutation counts ordered arrangements; a combination counts unordered selections. A president and vice president from ten: ninety. A two-person committee: forty-five. Ask whether swapping two chosen items makes a new outcome; if it does, order matters.

Which count to use, step by step.

ACT 1431 Permutation or Combination

Decide whether order matters, titles and rankings yes, teams and committees no; identify the pool and the number chosen; then compute the permutation, the combination, or a product across different pools. Gold, silver and bronze among eight runners: 336 ways. A relay pool of three with no roles: fifty-six. A sandwich and a drink from four and three: twelve. The combination is the permutation divided by r factorial, so two choices differing by that factor turn on whether order matters.

Two kinds of sequence.

ACT 1432 Arithmetic and Geometric Sequences

An arithmetic sequence adds a common difference, and a geometric sequence multiplies by a common ratio. Five, eight, eleven adds three; three, six, twelve doubles. Both formulas use n minus one, because the first term has taken no steps yet.

Finding a term, step by step.

ACT 1433 Finding a Term of a Sequence

Decide whether the sequence adds or multiplies, find the first term and the difference or ratio, and count n minus one steps. Seven, eleven, fifteen has a fiftieth term of seven plus forty-nine times four, 203. Two, six, eighteen has an eighth term of two times three to the seventh, 4,374. With a third term of ten and a seventh of twenty-six, four steps add sixteen, so the difference is four. From the third term to the seventh is four steps, not five.

The whole unit in one data set.

ACT 1434 One Class, Four Questions

Statistics and probability ask four questions: how the data were collected, what they look like, how two variables relate, and how likely an event is. A teacher's nine quiz scores run from sixty-two to ninety-seven: the mean is about 78.7, the median seventy-eight, the quartiles seventy-two and eighty-six, the interquartile range fourteen, and no score lies more than twenty-one points beyond a quartile, so there are no outliers. A tenth student scoring forty moves the median to 76.5 but drops the mean to 74.8. Choosing two of the original nine to present gives thirty-six pairs. Order data before finding medians, choose a probability's denominator from the given condition, and decide whether order matters before counting. This ends the mathematics.

Look at what you can now do. You can find probabilities with and, multiplying for independent events, and with or, subtracting the overlap. You can find the chance of at least one as one minus the chance of none. You can choose a conditional probability's denominator from the given condition, and find an expected value. You can count with permutations when order matters and combinations when it does not. And you can find a term of a sequence by counting its steps. Unit sixteen begins the reading section, with what a passage says and what it implies.