Compute theoretical and experimental probabilities, apply the addition rule for either-or events and the multiplication rule for independent events, and find complements.

What this topic is asking

The Statistics and Probability category includes probability (the S-CP standards). On the Grade 10 MCAS you compute theoretical and experimental probabilities, use the complement rule, apply the addition rule for either-or events, and use the multiplication rule for independent events. The decisions, when to add and when to multiply, are exactly the reasoning the test checks, so understanding the structure of the question matters most.

Theoretical and experimental probability

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). For equally likely outcomes:

For a bag of 3 red, 5 blue, 2 green marbles, $P(\text{blue}) = \frac{5}{10} = \frac{1}{2}$. This is theoretical probability, computed from the known model.

Experimental probability comes from actually performing trials: if 200 flips of a coin give 108 heads, the experimental $P(\text{heads}) = \frac{108}{200} = 0.54$. Over many trials, experimental probability tends toward the theoretical value (the law of large numbers), but in a small sample it can differ.

The complement rule

The complement of an event is its not-happening. Because every outcome either is or is not in the event:

This is often the fast route to "at least one" problems. If $P(\text{at least one defective})$ is wanted and $P(\text{none defective}) = 0.7$, then $P(\text{at least one}) = 1 - 0.7 = 0.3$. Computing the complement and subtracting is usually easier than adding many cases.

The addition rule (or)

For the probability that $A$ or $B$ happens, the addition rule is:

The subtraction avoids double-counting outcomes in both events. When $A$ and $B$ are mutually exclusive (cannot happen together, so $P(A \text{ and } B) = 0$), it simplifies to $P(A) + P(B)$. Drawing a card that is "a king or a queen" uses the simple form (a card cannot be both); "a king or a heart" needs the subtraction (the king of hearts is in both).

The multiplication rule (and, independent)

For two independent events (one does not affect the other's probability), the probability that both happen is the product:

Coin flips, spinner spins, and draws with replacement are independent. Draws without replacement are not, because removing one item changes the next probability. The MCAS keeps most Grade 10 multiplication-rule items to independent cases, but reading whether the first outcome changes the second is the key check.

Try this

Q1. A die is rolled. What is $P(\text{not a 6})$?

• Cue. $1 - \frac{1}{6} = \frac{5}{6}$.

Q2. Two coins are flipped. What is $P(\text{both tails})$?

• Cue. $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.