Theoretical Probability of Simple Events

Q: Can a probability be greater than 1 or negative?

No. A probability is always between $0$ and $1$, including both ends. If your fraction comes out bigger than $1$, you counted more favorable outcomes than total outcomes — recount. If it comes out negative, a subtraction went wrong.

Q: Should I write probability as a fraction, decimal, or percent?

All three are valid — $\dfrac{1}{4}$, $0.25$, and $25\%$ mean the same thing. In pre-algebra, give a reduced fraction unless the problem asks otherwise; multiple-choice answers are almost always reduced fractions.

Probability measures how likely something is to happen, as a number from $0$ to $1$ . A probability of $0$ means the event is impossible, $1$ means it is certain, and $\dfrac{1}{2}$ means it happens half the time. Rolling a number cube, spinning a spinner, drawing a marble from a bag — each one is an experiment, and probability predicts its results.

For the simple events in pre-algebra, one formula does all the work: count the outcomes you want, count all the outcomes possible, and divide. Almost every mistake on this topic is a counting mistake, not a math mistake — so the skill is really careful counting.

The formula

The probability of an event is the number of favorable outcomes divided by the total number of outcomes: $P(\text{event}) = \dfrac{\text{favorable outcomes}}{\text{total outcomes}}$ . Favorable just means the outcomes that count as the event you are asking about.

This formula only works when every outcome is equally likely — each face of a number cube, each equal section of a spinner, each marble in the bag. That is why problems always say equal sections or drawn at random.

Write the answer as a fraction and reduce it. $P = \dfrac{2}{6}$ should be reported as $\dfrac{1}{3}$ . A probability can never be less than $0$ or greater than $1$ — if you get $\dfrac{7}{6}$ , something got counted twice.

Counting outcomes carefully

Start by listing the whole sample space — every outcome that could happen. For one roll of a standard number cube that is $1, 2, 3, 4, 5, 6$ , so the total is $6$ . Then go through the list and mark which outcomes fit the event. Less than $4$ means $1, 2, 3$ — three outcomes, and $4$ itself is not one of them. At least $4$ means $4, 5, 6$ . Read those boundary words slowly; they are where the counting errors live.

For a bag of marbles, the total is all the marbles added together, not the number of colors. A bag with $3$ red, $5$ blue, and $2$ green marbles has $10$ equally likely outcomes, so $P(\text{red}) = \dfrac{3}{10}$ .

The complement: probability of "not"

Every event either happens or it does not, so the probabilities of an event and its opposite add to $1$ : $P(\text{not A}) = 1 - P(\text{A})$ . If the probability of drawing a green marble is $\dfrac{1}{5}$ , the probability of not drawing green is $\dfrac{4}{5}$ .

You can also find a not probability directly by counting everything except the event. Both routes give the same answer, and computing it both ways is a fast built-in check.

Worked examples

Example 1: rolling a number cube

A standard number cube is rolled once. What is the probability of rolling an even number?

List the sample space

1, 2, 3, 4, 5, 6

Count the favorable outcomes — the evens

2, 4, 6 \text{ — that is } 3 \text{ outcomes}

Apply the formula

P = \dfrac{3}{6}

Reduce

P = \dfrac{1}{2}

Answer: $P(\text{even}) = \dfrac{1}{2}$

Example 2: drawing a marble

A bag holds $3$ red, $5$ blue, and $2$ green marbles. One marble is drawn at random. What is the probability it is red?

Count the total outcomes

3 + 5 + 2 = 10

Count the favorable outcomes

3 \text{ red}

Apply the formula

P(\text{red}) = \dfrac{3}{10}

Answer: $P(\text{red}) = \dfrac{3}{10}$

Example 3: a spinner

A spinner has $8$ equal sections numbered $1$ – $8$ . What is the probability of spinning a number greater than $6$ ?

Total outcomes

8

Favorable outcomes — greater than

6

, so

6

itself does not count

7, 8 \text{ — that is } 2 \text{ outcomes}

Apply the formula

P = \dfrac{2}{8}

Reduce

P = \dfrac{1}{4}

Answer: $P = \dfrac{1}{4}$

Example 4: the complement

Using the same bag of $3$ red, $5$ blue, and $2$ green marbles, what is the probability the marble drawn is not green?

Find the probability of green

P(\text{green}) = \dfrac{2}{10} = \dfrac{1}{5}

Subtract from

1

P(\text{not green}) = 1 - \dfrac{1}{5} = \dfrac{4}{5}

Check by counting directly:

3 + 5 = 8

non-green marbles, and

\dfrac{8}{10} = \dfrac{4}{5}

✓

Answer: $P(\text{not green}) = \dfrac{4}{5}$

Try one yourself

Common questions

Can a probability be greater than 1 or negative?

No. A probability is always between $0$ and $1$ , including both ends. If your fraction comes out bigger than $1$ , you counted more favorable outcomes than total outcomes — recount. If it comes out negative, a subtraction went wrong.

Should I write probability as a fraction, decimal, or percent?

All three are valid — $\dfrac{1}{4}$ , $0.25$ , and $25\%$ mean the same thing. In pre-algebra, give a reduced fraction unless the problem asks otherwise; multiple-choice answers are almost always reduced fractions.

What is the difference between theoretical and experimental probability?

Theoretical probability comes from counting outcomes before you do anything — a fair coin gives heads with probability $\dfrac{1}{2}$ . Experimental probability comes from actual results: if you flip $50$ times and get $28$ heads, the experimental probability is $\dfrac{28}{50} = \dfrac{14}{25}$ . With more trials, the experimental value tends to settle near the theoretical one.

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Theoretical Probability of Simple Events

The formula

Counting outcomes carefully

The complement: probability of "not"

Worked examples

Example 1: rolling a number cube

Example 2: drawing a marble

Example 3: a spinner

Example 4: the complement

Try one yourself

Common questions

Keep learning

Measures of Center (Mean, Median, Mode)

Percent of a Number

Unit Rate as Slope