Topic 4.3 Introduction to Probability: apply the basic properties of probability (range, total of one, complement rule) and the law of large numbers to compute and interpret probabilities of events.

What this topic is asking

The College Board (Topic 4.3) wants you to apply the basic properties of probability, the valid range, the total-of-one rule, and the complement rule, alongside the law of large numbers, to compute and interpret probabilities of events.

The basic properties

These are the ground rules every probability obeys, and the exam tests them directly by giving a list of probabilities and asking for a missing one (use total-of-one), or by offering an impossible value like $1.3$ or $-0.2$ (violates the range). Internalising "no probability is below $0$ or above $1$, and everything sums to $1$" catches a surprising number of errors.

The complement rule

The complement rule is worth its own emphasis because so many questions are far easier through it. "The probability of at least one success" is almost always best computed as $1 - P(\text{no successes})$, since "no successes" is one simple case while "at least one" is many. Trained to spot "at least," "not," and "none," you will reach for the complement automatically.

Equally likely outcomes and the law of large numbers

When a sample space has equally likely outcomes (a fair die, a well-shuffled deck), probability reduces to counting: the probability of an event is the count of favorable outcomes over the total count. So a fair die has $P(\text{even}) = 3/6 = 0.5$. This counting definition is exact and needs no experiment. But not every process has equally likely outcomes (a bent coin, an unequal spinner), and for those the probability is the long-run relative frequency that the law of large numbers describes: run the process many times and the proportion of the event settles toward its probability. The two views fit together. Where outcomes are equally likely, the counting probability is the value the long-run proportion approaches; where they are not, the long-run proportion (or a model) supplies the probability that counting cannot. Holding both pictures, theoretical counting and empirical long-run frequency, lets you handle fair and unfair processes alike, and it explains why simulation (Topic 4.2) is a legitimate way to estimate any probability.

Why these rules anchor the unit

The properties in Topic 4.3 look elementary, but they are the axioms on which every later rule is built. The addition rule, the multiplication rule, conditional probability, and the distributions of Unit 4 all reduce, ultimately, to "probabilities are numbers in $[0,1]$ that sum to $1$ over the sample space." When a later calculation gives a probability above $1$ or a set of probabilities that do not sum to $1$, these axioms are your error check: the answer must be wrong. Likewise, the complement rule reappears constantly, in binomial "at least one" problems, in geometric "first success by trial $k$" problems, and throughout inference. So treating these basics as load-bearing, not trivial, pays off: a firm grip on the range, the total-of-one rule, and the complement makes the harder rules feel like natural extensions rather than new facts to memorize.

Try this

Q1. If $P(A) = 0.62$, find $P(A^c)$ and explain the rule used. [2 points]

• Cue. $P(A^c) = 1 - 0.62 = 0.38$, by the complement rule, because $A$ and "not $A$" together are certain and cannot both happen.

Q2. A student reports $P(\text{event}) = 1.2$. Explain why this must be an error. [1 point]

• Cue. Every probability satisfies $0 \le P \le 1$; a value above $1$ is impossible because nothing is more than certain.