Experimental Probability and the Long Run | Grade 7 Math | LevelKite

You already know how to find a probability by counting. A fair coin has two equally likely outcomes, heads and tails, so the probability of heads is $\frac{1}{2}$ .

Some chances can't be counted that way. Toss a bottle cap. It can land open side up or open side down, but you have no reason to think those two results are equally likely. Looking at the cap won't tell you the chance. You have to toss it again and again and keep track of what happens. Repeating a chance action and recording the results is called an experiment.

In this lesson, you'll turn the results of an experiment into a probability. You'll also see why the number of tries matters so much.

These words help you describe an experiment.

A trial is one run of the experiment: one toss, one spin or one roll.
An event is the result you are watching for, such as "the cap lands open side up." An event can be one outcome or a group of outcomes.
The frequency of an event is the number of trials in which the event happened.
The relative frequency of an event is its frequency compared with the total number of trials, written as the fraction $\frac{\text{frequency}}{\text{total trials}}$ .

The experimental probability of an event is its relative frequency in the trials you actually ran. It answers the question: out of all my trials, what fraction went this way?

One bottle cap tossed 50 times

Event	Frequency (number of tosses)	Total trials (tosses)	Relative frequency (frequency ÷ total trials)	As a decimal	As a percent
Cap lands open side up	$23$	$50$	$\frac{23}{50}$	$\frac{23}{50} = \frac{46}{100} = 0.46$	$46\%$
Cap lands open side down	$27$	$50$	$\frac{27}{50}$	$\frac{27}{50} = \frac{54}{100} = 0.54$	$54\%$
Both events together	$23 + 27 = 50$	$50$	$\frac{50}{50}$	$0.46 + 0.54 = 1.00$	$46\% + 54\% = 100\%$

Read the first row from left to right. The cap landed open side up in $23$ tosses, so the frequency is $23$ . There were $50$ tosses in all, so the relative frequency is $\frac{23}{50}$ . This fraction compares a part (the tosses that went this way) with the whole (every toss). That is why you divide by the total number of trials, $50$ , and not by the $27$ tosses that went the other way.

To write $\frac{23}{50}$ as a decimal, multiply the numerator and the denominator by $2$ . Multiplying both by the same nonzero number does not change the value of a fraction, so $\frac{23}{50} = \frac{46}{100} = 0.46$ . Percent means "out of $100$ ," so this is also $46\%$ .

Now look at the last row. In this experiment, every toss landed one way or the other and never both, so the two frequencies add up to the total: $23 + 27 = 50$ . That is why the two relative frequencies add up to $1$ : $0.46 + 0.54 = 1.00$ . You can use this as a check whenever your events cover every trial exactly once.

Key concept

Experimental probability is relative frequency

The experimental probability of an event is the fraction of your trials in which the event happened. You need at least one trial. The result is always from $0$ to $1$ , because the frequency can't be less than $0$ or more than the total number of trials.

This number describes the trials you ran. Over many trials, it approximates the probability of the event, which means it comes close to it. Over only a few trials, it may not. Even with many trials, it is an estimate, not a guarantee.

\text{experimental probability} = \frac{\text{number of trials in which the event happened}}{\text{total number of trials}}

Worked example

One outcome, total given

You spin a spinner $40$ times. It lands on blue $14$ times. What is the experimental probability of landing on blue?

Name the event and the two counts you need. The event is "the spinner lands on blue." Its frequency is 14, the number of spins that landed on blue. The total number of trials is 40, because each spin is one trial.
$\text{frequency} = 14, \quad \text{total trials} = 40$
Write the relative frequency. Put the frequency over the total number of trials, because experimental probability compares the spins that landed on blue with all of the spins.
$\frac{14}{40}$
Simplify the fraction. Both 14 and 40 are divisible by 2, and dividing the numerator and the denominator by the same nonzero number keeps the value of the fraction the same.
$\frac{14}{40} = \frac{14 \div 2}{40 \div 2} = \frac{7}{20}$
Write the result as a decimal and a percent. Multiply the numerator and the denominator by 5 to get a denominator of 100, which also keeps the value the same. Thirty-five hundredths is 0.35, which is 35%.
$\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100} = 0.35 = 35\%$

Answer

In these $40$ spins, the experimental probability of landing on blue is $\frac{7}{20}$ , which is $0.35$ or $35\%$ .

Worked example

A group of outcomes, total not given

You roll a fair number cube, with faces numbered $1$ to $6$ , and record how many times each number comes up: $1$ comes up $9$ times, $2$ comes up $11$ times, $3$ comes up $8$ times, $4$ comes up $14$ times, $5$ comes up $7$ times and $6$ comes up $11$ times. What is the experimental probability of rolling a number greater than $4$ ?

Find the total number of trials. It isn't given, but each roll shows exactly one number, so every roll is counted exactly once in the list. Add all six frequencies, two at a time.
$9 + 11 + 8 + 14 + 7 + 11 = 20 + 22 + 18 = 60$
Find the frequency of the event. The numbers greater than 4 on the cube are 5 and 6. One roll can't show both, so adding their two frequencies counts each of these rolls exactly once.
$7 + 11 = 18$
Write the relative frequency: the frequency of the event over the total number of trials.
$\frac{18}{60}$
Simplify by dividing the numerator and the denominator by 6, their greatest common factor, which keeps the value the same. Three tenths is 0.3, the same as 30 hundredths, so it is 30%.
$\frac{18}{60} = \frac{18 \div 6}{60 \div 6} = \frac{3}{10} = 0.3 = 30\%$

Answer

In these $60$ rolls, the experimental probability of rolling a number greater than $4$ is $\frac{3}{10}$ , which is $0.3$ or $30\%$ .

Look back at the number cube. On a fair cube, each of the $6$ numbers is equally likely, and $2$ of them are greater than $4$ . So the probability of rolling a number greater than $4$ is $\frac{2}{6} = \frac{1}{3}$ , or about $0.33$ . Yet your experiment gave $0.3$ . Did something go wrong?

No. An experimental probability depends on what happened to come up in your trials, so it often doesn't match the probability exactly. Roll another $60$ times and you will probably get a different value.

How close it is likely to be depends on the number of trials. With only a few trials, the relative frequency can swing a lot. As you keep adding trials, it tends to settle near one value. The relative frequency over many trials is called the long-run relative frequency. It gives you a good approximation of the probability of the event, but not a guaranteed exact value.

You can watch this happen with a fair coin, because you know its probability of heads is $\frac{1}{2} = 0.5$ . The table shows one coin flipped $1000$ times, with the totals checked along the way.

One fair coin flipped 1000 times, checked along the way

Flips so far (trials)	Heads so far (frequency)	Relative frequency (heads ÷ flips)	As a decimal	Distance from 0.5
$10$	$6$	$\frac{6}{10}$	$\frac{6}{10} = 0.60$	$0.60 - 0.50 = 0.10$
$20$	$13$	$\frac{13}{20}$	$\frac{13}{20} = \frac{65}{100} = 0.65$	$0.65 - 0.50 = 0.15$
$50$	$29$	$\frac{29}{50}$	$\frac{29}{50} = \frac{58}{100} = 0.58$	$0.58 - 0.50 = 0.08$
$100$	$53$	$\frac{53}{100}$	$\frac{53}{100} = 0.53$	$0.53 - 0.50 = 0.03$
$500$	$244$	$\frac{244}{500}$	$\frac{244}{500} = \frac{488}{1000} = 0.488$	$0.500 - 0.488 = 0.012$
$1000$	$506$	$\frac{506}{1000}$	$\frac{506}{1000} = 0.506$	$0.506 - 0.500 = 0.006$

Each row uses the same rule as before: heads so far divided by flips so far, then rewritten as a decimal.

Read down the decimals: $0.60$ , $0.65$ , $0.58$ , $0.53$ , $0.488$ , $0.506$ . In the early rows, the relative frequency jumps around. From $10$ flips to $20$ flips it even moves farther from $0.5$ . That can happen, because more trials do not promise a closer result at every step.

The last column shows how far each relative frequency is from $0.5$ . A distance is never negative, so you subtract the smaller number from the larger one. After $20$ flips the distance is $0.15$ . After $1000$ flips it is only $0.006$ .

Why does the swinging calm down? Each flip becomes a smaller share of a bigger total. One flip is $\frac{1}{10}$ of the first $10$ flips, so it can shift the relative frequency a lot. One flip is only $\frac{1}{1000}$ of $1000$ flips, so it barely shifts it. That is why, over many trials, the relative frequency tends to settle near the probability. It is still an approximation: $0.506$ is close to $0.5$ , not equal to it.

Worked example

A probability you can't find by counting

Your class keeps tossing the bottle cap from the start of the lesson and tracks the totals so far. After $50$ tosses, the cap has landed open side up $23$ times. After $200$ tosses, the count is $82$ . After $800$ tosses, the count is $304$ . Approximate the probability that this cap lands open side up.

Find the relative frequency after 50 tosses: the open-side-up count divided by the tosses so far. Multiply the numerator and the denominator by 2 to get hundredths, which keeps the value the same.
$\frac{23}{50} = \frac{23 \times 2}{50 \times 2} = \frac{46}{100} = 0.46$
Do the same after 200 tosses. This time, divide the numerator and the denominator by 2 to get hundredths.
$\frac{82}{200} = \frac{82 \div 2}{200 \div 2} = \frac{41}{100} = 0.41$
Do the same after 800 tosses. Divide the numerator and the denominator by 8 to get hundredths.
$\frac{304}{800} = \frac{304 \div 8}{800 \div 8} = \frac{38}{100} = 0.38$
Compare the three values. From 50 to 200 tosses, the relative frequency dropped by 0.05. From 200 to 800 tosses, with 600 more tosses added, it dropped by only 0.03. The swings are getting smaller.
$0.46 - 0.41 = 0.05 \qquad 0.41 - 0.38 = 0.03$
Choose the value from the most trials. The 800 tosses include all the earlier tosses plus many more, and a relative frequency from more trials tends to be closer to the probability. The symbol ≈ means "is approximately equal to."
$\text{probability of open side up} \approx 0.38$

Answer

The probability that this cap lands open side up is about $0.38$ , or about $38\%$ . This is an approximation based on $800$ tosses, not an exact value.

Here is the idea to keep. The experimental probability of an event is its relative frequency: the number of trials in which the event happened, divided by the total number of trials. It describes the trials you ran.

To approximate the probability of an event, use the relative frequency from many trials. With only a few trials, the result can swing widely. Over many trials, it tends to settle. It stays an estimate. It can't tell you what the next trial will be, and any count you expect from it is approximate, never guaranteed.

Next, you'll go the other way. You'll start with a probability and predict about how often an event should happen.