HiSET · Mathematics · 5 min read
Probability and Algebraic Expressions
- Calculate theoretical, experimental, and compound probabilities using counting rules and probability formulas.
- Translate written phrases into accurate algebraic expressions using variables, operational symbols, and grouping signs.
- Evaluate algebraic expressions correctly by substituting values and applying the standard order of operations.
Understanding Outcomes and Basic Probability
In probability, the sample space is the set of all possible outcomes of an experiment. When determining the total number of outcomes across combined choices, we multiply the possibilities. If a first choice has m possible outcomes and a second choice has n possible outcomes for each first outcome, there are m × n possible combined outcomes. Similarly, if three successive choices have a, b, and c possible outcomes at every stage, there are a × b × c possible combined outcomes. To see all possible paths clearly, a tree diagram can be drawn, or an organized list or grid can systematically record every possible outcome without omission or duplication.
The theoretical probability of an event is the ratio of favorable outcomes to the total number of equally likely outcomes. The probability of any event is always a number from 0 to 1, inclusive. A probability of 0 indicates that an event cannot occur, while a probability of 1 indicates that an event is certain to occur. You can represent probabilities equivalently as fractions, decimals, or percentages.
The complement of event A consists of all possible outcomes in the sample space where event A does not occur. Because an event either happens or does not happen, the probability of the complement of event A equals 1 minus the probability of event A. Furthermore, the sum of the probabilities of all mutually exclusive outcomes in a complete sample space equals 1.
While theoretical probability calculates what should happen mathematically, experimental probability is the ratio of the number of times an outcome occurs to the total number of trials conducted in an actual test. As the total number of experimental trials increases, the experimental probability typically gets closer to the theoretical probability.
| Probability Concept | Definition | Value Range or Calculation |
|---|---|---|
| Theoretical Probability | Ratio of favorable outcomes to total equally likely outcomes | Between 0 and 1, inclusive |
| Experimental Probability | Ratio of times an outcome occurs to total trials conducted | Between 0 and 1, inclusive |
| Complement of Event A | All outcomes in the sample space where event A does not occur | 1 minus the probability of event A |
| Certain Event | An event guaranteed to occur | Exactly 1 (or 100%) |
| Impossible Event | An event that cannot occur | Exactly 0 (or 0%) |
Common mistakes
- Reversing subtraction order: Writing 5 - x instead of x - 5 when translating the phrase '5 less than x'.
- Double-counting overlapping outcomes: Forgetting to subtract the probability of both events occurring when calculating the probability of A or B for non-mutually exclusive events.
- Misapplying exponents to negative signs: Confusing -x² with (-x)², forgetting that without parentheses the exponent applies before the negative sign.
- Treating dependent events as independent: Failing to reduce the total number of items in the sample space when items are chosen without replacement.
Key points
- The total number of combined outcomes for successive stages is found by multiplying the number of choices at each stage (m × n or a × b × c).
- Probabilities range from 0 (impossible) to 1 (certain) and can be written as fractions, decimals, or percentages.
- The probability of the complement of event A is 1 minus the probability of event A, and the sum of all mutually exclusive outcomes in a complete sample space is 1.
- Multiply individual probabilities for independent events occurring together; add individual probabilities and subtract overlap for non-mutually exclusive 'or' events.
- Independent random draws with replacement keep the pool unchanged; without replacement, the pool changes and events such as drawing the same color twice are generally dependent.
- Phrases such as 'less than' and 'subtracted from' reverse the written order of terms in algebraic subtraction.
- When evaluating expressions, follow the order of operations: grouping symbols first, then exponents/roots, followed by multiplication and division from left to right, and finally addition and subtraction from left to right.
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Sample lesson: Rational Numbers and Arithmetic Operations25 practice questions on this exam
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