Key takeaway: Probability is the measure of how likely an event is, placed on a scale where 0 stands for an impossible event and 1 stands for a sure thing.
Grade 77.SP.C.5Probability and Counting
Definition of Probability
Probability puts a number on chance. Instead of saying that rain is “pretty likely,” a forecaster says the chance is 0.8, which also reads as 80%. The scale starts at 0 for events that cannot happen and ends at 1 for events that must happen. Everything in between is some degree of likelihood.
When all outcomes are equally likely, count the outcomes you want and divide by every outcome that could occur. A result below 0 or above 1 signals an error, so the scale doubles as a built-in check on your work.
Formula and notation
The probability of an event \(E\) is written \(P(E)\):
\[P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}, \qquad 0 \le P(E) \le 1\]
The chance that the event does not happen is \(1 – P(E)\).
Key facts
- Rolling a 7 on one standard die is impossible, so its chance is 0.
- A value of \(\frac{1}{2}\) means happening and not happening are equally likely.
- The chance of an event and the chance of its opposite always sum to 1.
- The same value can be written as a fraction, a decimal or a percent.
Probability example
A bag holds 15 marbles: 6 red, 4 blue and 5 green. One marble is drawn without looking. Find the probability of blue and of not red.
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Blue: 4 favorable outcomes out of 15, so \(P(\text{blue}) = \frac{4}{15} \approx 0.27\).
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Not red means blue or green: \(4 + 5 = 9\) outcomes.
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\(P(\text{not red}) = \frac{9}{15} = \frac{3}{5}\). Check: \(1 – \frac{6}{15} = \frac{9}{15}\).
\(P(\text{blue}) = \) \(\frac{4}{15}\) and \(P(\text{not red}) = \) \(\frac{3}{5}\).
Common mistake
Watch out
Hector writes \(P(\text{blue}) = \frac{4}{11}\), because there are 4 blue marbles and 11 marbles that are not blue.
Why it is wrong
He compared favorable outcomes with the unfavorable ones. That is the idea of odds. The denominator must be all possible outcomes, including the favorable ones: \(4 + 11 = 15\). So the correct value is \(\frac{4}{15}\).
Related terms and lessons
- Theoretical Probability
- Experimental Probability
- Sample Space
- Compound Event
- Conditional Probability
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