Can a probability be greater than 1 or negative?

No. The favorable outcomes are always part of the total outcomes, so the fraction can never exceed totaltotal = 1, and you can never count a negative number of outcomes. If your answer is above 1 or below 0, you probably flipped the fraction or subtracted in the wrong order. Percent versions run from 0% to 100%.

What is the difference between an outcome and an event?

An outcome is one single result, like rolling a 3. An event can include several outcomes, like "rolling an odd number," which covers 1, 3, and 5. To find the probability of an event, count every outcome that makes it happen and put that count over the total number of equally likely outcomes.

If the probability of rain is 40%, does that mean it will rain?

No. Probability describes long-run expectation, not a guarantee about one day. A 40% chance means that on days with these conditions, rain happens about 40 out of every 100 times. It might rain or it might not; the complement tells you there is a 60% chance of no rain, which is the more likely outcome but still not certain.

Which form should I use for my answer — fraction, decimal, or percent?

Give whatever form the problem asks for, and use the exact fraction as your foundation. Fractions are exact even when the decimal repeats, decimals make comparing two probabilities easy, and percents are best for describing chance in everyday language. If you convert, round the decimal only at the very end and write "about" when the decimal doesn't terminate.

M7MATH-10.1

Probability of Simple Events

Learn how to find the probability of a simple event, place it on the 0-to-1 scale, use the complement rule, and convert answers to fractions, decimals, and percents.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Probability of Simple Events, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Probability of Simple Events free

What this lesson covers

Every time you spin a spinner, roll a number cube, or pull a marble out of a bag without looking, you are running a chance experiment. Probability is the number that tells you how likely a particular result is — not what will happen, but how often you'd expect it to happen over the long run.

In this lesson you'll learn to count favorable outcomes and total equally likely outcomes, write the probability as a fraction, and then translate that fraction into a decimal and a percent. You'll also place probabilities on the 0-to-1 scale so you can describe an event as impossible, unlikely, as likely as not, likely, or certain. Finally, you'll use the complement rule — a shortcut that lets you find the chance an event does not happen by subtracting from 1.

Outcomes, Events, and the Basic Formula

A chance experiment is any action with a result you can't predict for sure, like rolling a number cube. Each possible result is an outcome, and the full list of outcomes is the sample space. Rolling a standard number cube has sample space

1, 2, 3, 4, 5, 6

— six outcomes.

An event is the outcome or group of outcomes you care about. "Rolling an even number" is an event made of three outcomes: 2, 4, and 6. Outcomes that make the event happen are called favorable outcomes — favorable doesn't mean good, just "counted."

When every outcome is equally likely, probability is

P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{number of equally likely outcomes}}

P(\text{even}) = \frac{3}{6} = \frac{1}{2}

.

That phrase "equally likely" matters. If a bag holds 3 red and 5 blue marbles and you draw without looking, the eight marbles are equally likely, but the two colors are not. The denominator is 8, not 2. A very common error is writing

P(\text{red}) = \frac{1}{2}

because there are two colors. Always count individual outcomes, never categories.

Another frequent slip is putting the total on top. The favorable count can never be larger than the total, so a correct probability of a simple event is always a number from 0 to 1. If you ever get

\frac{8}{3}

, you flipped the fraction.

The 0-to-1 Probability Scale

Every probability lives on a number line from 0 to 1. Nothing is ever negative, and nothing is ever more than 1.

Probability	Decimal	Percent	Description
$0$	0	0%	impossible
near $0$ , like $\frac{1}{20}$	0.05	5%	very unlikely
$\frac{1}{4}$	0.25	25%	unlikely
$\frac{1}{2}$	0.5	50%	as likely as not
$\frac{3}{4}$	0.75	75%	likely
$1$	1	100%	certain

A probability of 0 means the event cannot happen: rolling a 9 on a standard number cube. A probability of 1 means it must happen: rolling a number less than 10. Anything in between is a matter of degree, and the closer to 1, the more likely.

To compare two events, get them into the same form. Is

\frac{3}{8}

or 0.4 more likely? Convert:

\frac{3}{8} = 0.375

, which is less than 0.4, so the 0.4 event is more likely. Comparing a fraction to a percent by eyeballing is where students usually go wrong.

When a teacher asks you to describe an event in words, anchor to the benchmarks

0

\frac{1}{4}

\frac{1}{2}

\frac{3}{4}

, and

1

. An event with probability

\frac{7}{8} = 0.875

is close to 1, so "likely" — almost certain, but not certain, because there is still one outcome that would make it fail.

Fraction, Decimal, and Percent Forms

The same probability can be written three ways, and you should be fluent moving among them.

Start from the fraction, since that's what counting gives you. Simplify it if you can:

\frac{4}{10} = \frac{2}{5}

. To get the decimal, divide the numerator by the denominator:

2 \div 5 = 0.4

. To get the percent, multiply the decimal by 100 and attach a percent sign:

0.4 \times 100 = 40\%

.

Going backward works too. A 65% chance is

0.65

, which is

\frac{65}{100} = \frac{13}{20}

.

Some probabilities don't give terminating decimals.

P = \frac{1}{3} = 0.333\ldots \approx 0.33

, or about 33%. When that happens, keep the exact fraction as your main answer and round the decimal, saying "about." Writing

\frac{1}{3} = 0.3

is wrong, and rounding too early then comparing two events can flip your conclusion.

Fraction	Decimal	Percent
$\frac{1}{8}$	0.125	12.5%
$\frac{1}{5}$	0.2	20%
$\frac{1}{6}$	$\approx 0.167$	$\approx 16.7\%$
$\frac{5}{6}$	$\approx 0.833$	$\approx 83.3\%$

One more caution: a percent is not the same as a count. If 25% of spins land on blue, that does not mean 25 spins land on blue. Percent tells you the rate you'd expect, and you'd multiply by the number of trials to predict a count.

The Complement Rule

The complement of an event is everything in the sample space that is not that event. If event

A

is "drawing a red marble," then the complement, written

A'

or "not

A

," is "drawing a marble that isn't red."

Because every outcome either is in

A

or isn't, the two probabilities must add to the whole:

P(A) + P(\text{not } A) = 1 \qquad \text{so} \qquad P(\text{not } A) = 1 - P(A)

This is a genuine time-saver. A bag has 20 tiles: 3 are labeled with a star. Rather than counting all 17 non-star tiles, compute

P(\text{star}) = \frac{3}{20}

and then

P(\text{not star}) = 1 - \frac{3}{20} = \frac{17}{20}

. Same answer, less counting — and it's especially useful with "at least one" wording later in the unit.

In percent form it's even quicker: if rain is 30% likely, no rain is

100\% - 30\% = 70\%

likely.

Two mistakes show up over and over. First, students subtract from the total number of outcomes instead of from 1, writing

20 - \frac{3}{20}

. The complement rule subtracts probabilities from 1, always. Second, students think the complement of "rolling a 2" is "rolling a 5," as if it were the opposite side. The complement of "rolling a 2" is "rolling a 1, 3, 4, 5, or 6" — everything else, all at once, with probability

1 - \frac{1}{6} = \frac{5}{6}

Key terms

Outcome.: One possible result of a chance experiment, such as landing on 4 when you roll a number cube.
Sample space.: The complete list of all possible outcomes of a chance experiment.
Event.: An outcome or set of outcomes you are interested in, such as "rolling an odd number."
Favorable outcome.: An outcome that belongs to the event being counted; it goes in the numerator of the probability.
Equally likely.: Describes outcomes that each have the same chance of occurring, which is what lets you use the favorable-over-total formula.
Probability of a simple event.: The number of favorable outcomes divided by the number of equally likely outcomes, always between 0 and 1.
Complement.: All outcomes in the sample space that are not in the event; its probability is $1 - P(\text{event})$ .
Certain / impossible.: An event with probability 1 must happen; an event with probability 0 cannot happen.

Worked example

A spinner is divided into 8 equal sections: 3 green, 2 yellow, 1 purple, and 2 orange. The spinner is spun once. (a) Find

P(\text{yellow})

as a fraction, decimal, and percent. (b) Find

P(\text{not green})

using the complement rule. (c) Describe the likelihood of spinning green or orange in words.

Step 1: Check that the outcomes are equally likely. The 8 sections are equal in size, so each section is equally likely. The total number of equally likely outcomes is 8.

Step 2: Part (a), count favorable outcomes for yellow. There are 2 yellow sections, so

P(\text{yellow}) = \frac{2}{8} = \frac{1}{4}

Decimal:

1 \div 4 = 0.25

. Percent:

0.25 \times 100 = 25\%

. So

P(\text{yellow}) = \frac{1}{4} = 0.25 = 25\%

.

Step 3: Part (b), find the probability of green first. There are 3 green sections, so

P(\text{green}) = \frac{3}{8}

.

Step 4: Apply the complement rule. Subtract from 1, not from 8:

P(\text{not green}) = 1 - \frac{3}{8} = \frac{8}{8} - \frac{3}{8} = \frac{5}{8}

Check by direct counting: the non-green sections are 2 yellow, 1 purple, and 2 orange, which is

2 + 1 + 2 = 5

sections out of 8. It matches. In decimal form,

\frac{5}{8} = 0.625

, or

62.5\%

.

Step 5: Part (c), combine green and orange. Green and orange together cover

3 + 2 = 5

sections, so

P = \frac{5}{8} = 0.625

. That is a little more than

\frac{1}{2}

but well below

\frac{3}{4}

, so the event is slightly more likely than not — you'd expect it a bit more often than half the time.

Practice questions

A bag contains 5 red, 4 blue, and 11 white tiles, all the same size. One tile is drawn without looking. What is the probability the tile is NOT white?

$\frac{11}{20}$
$\frac{9}{20}$
$\frac{9}{11}$
$\frac{1}{3}$

Answer: $\frac{9}{20}$

The total number of equally likely outcomes is

5 + 4 + 11 = 20

tiles. Since

P(\text{white}) = \frac{11}{20}

, the complement rule gives

P(\text{not white}) = 1 - \frac{11}{20} = \frac{9}{20}

. You can check by counting directly: the non-white tiles are the 5 red plus the 4 blue, which is 9 out of 20. The choice

\frac{11}{20}

is the probability of white itself, and

\frac{1}{3}

comes from the mistake of counting three colors instead of twenty tiles.

A number cube with faces 1 through 6 is rolled once. Find

P(\text{a number greater than 4})

and write it as a fraction, a decimal rounded to the hundredths place, and a percent. Then find the probability of the complement and describe that complement in words.

Answer: $P(\text{greater than }4) = \frac{2}{6} = \frac{1}{3} \approx 0.33 \approx 33\%$ ; the complement is "rolling 4 or less," with probability $\frac{2}{3} \approx 0.67 \approx 67\%$ .

Only 5 and 6 are greater than 4, so there are 2 favorable outcomes out of 6 equally likely outcomes:

\frac{2}{6}

, which simplifies to

\frac{1}{3}

. Dividing,

1 \div 3 = 0.333\ldots \approx 0.33

, or about 33%. Because

\frac{1}{3}

is a repeating decimal, keep the fraction as the exact answer and use "about" for the decimal and percent. The complement is everything else in the sample space: 1, 2, 3, or 4 — that is, "4 or less," not just "exactly 4." Its probability is

1 - \frac{1}{3} = \frac{2}{3} \approx 0.67

, about 67%, and the two probabilities add to 1.

Mia says that since a spinner has 4 colors on it, the probability of landing on blue must be

\frac{1}{4}

. Explain when Mia is right and when she is wrong, and give an example of a spinner where she is wrong.

Answer: Mia is right only if the four colored regions are all the same size (equally likely). If the regions are different sizes, the colors are not equally likely and you must compare equal-sized parts instead.

The favorable-over-total formula requires equally likely outcomes. On a spinner cut into 4 equal quarters — one per color — each color truly has probability

\frac{1}{4}

. But imagine a spinner divided into 8 equal eighths where 5 are blue, 1 is red, 1 is green, and 1 is yellow. There are still 4 colors, yet

P(\text{blue}) = \frac{5}{8} = 0.625

, not

\frac{1}{4}

. The number of categories never sets the denominator; the number of equally likely pieces does.

FAQ

Can a probability be greater than 1 or negative?: No. The favorable outcomes are always part of the total outcomes, so the fraction can never exceed $\frac{\text{total}}{\text{total}} = 1$ , and you can never count a negative number of outcomes. If your answer is above 1 or below 0, you probably flipped the fraction or subtracted in the wrong order. Percent versions run from 0% to 100%.
What is the difference between an outcome and an event?: An outcome is one single result, like rolling a 3. An event can include several outcomes, like "rolling an odd number," which covers 1, 3, and 5. To find the probability of an event, count every outcome that makes it happen and put that count over the total number of equally likely outcomes.
If the probability of rain is 40%, does that mean it will rain?: No. Probability describes long-run expectation, not a guarantee about one day. A 40% chance means that on days with these conditions, rain happens about 40 out of every 100 times. It might rain or it might not; the complement tells you there is a 60% chance of no rain, which is the more likely outcome but still not certain.
Which form should I use for my answer — fraction, decimal, or percent?: Give whatever form the problem asks for, and use the exact fraction as your foundation. Fractions are exact even when the decimal repeats, decimals make comparing two probabilities easy, and percents are best for describing chance in everyday language. If you convert, round the decimal only at the very end and write "about" when the decimal doesn't terminate.

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