Experimental and theoretical probability

Two probabilities for the same event: one counted from trials, one reasoned from the setup. The extra step is that the event covers several outcomes, so they get added before anything is divided.

01 Worked Example Video

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A short walkthrough explaining what you need to know and how to solve this question type lands here once it's recorded.

02 Recognize the Question

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

Experimental only

A die-like machine outputs 1 to 6. In 400 trials the counts were 61, 70, 66, 68, 72, 63.

Compute the experimental probability of a number less than 3, to the nearest thousandth.

Theoretical only

Same machine, outputs 1 to 6, assumed fair. The 400 trials are still on the page.

Compute the theoretical probability of a number less than 3, to the nearest thousandth.

Which is which

A fair eight-sided spinner is spun 40 times and lands on an even number 24 times.

What is the theoretical probability of an even number? Give a decimal.

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 The Rule

experimental = (trials in the event) ÷ (total trials) counted from what happened

theoretical = (outcomes in the event) ÷ (all outcomes) reasoned from a fair setup

When the event covers more than one outcome, add before you divide. “Greater than 5” on an eight-sided machine is three outcomes, so the theoretical probability is 3/8, and the experimental one needs those same three trial counts summed.

Common slips

(1) Using one outcome instead of the group. The event usually spans several. (2) Dividing the theoretical by the trial count. Theoretical never touches the data; it comes from the setup alone. (3) Expecting the two to match exactly. They rarely do, and part (c) is about that.

04 Practice — Step by Step

Add the group, then divide

A quality lab is testing a machine that randomly stamps one of the digits 1 through 8 onto each part. The lab ran the machine for 250 trials and recorded the following.

Outcome	1	2	3	4	5	6	7	8
Number of trials	28	35	31	29	33	30	34	30

(a) From these results, compute the experimental probability of stamping a number greater than 5.
(b) Assuming the machine is fair, compute the theoretical probability of a number greater than 5.
(c) Choose the true statement.

Round to the nearest thousandth.

Step 1

Which outcomes count?

The event is “greater than 5” on a machine stamping 1 to 8. Which outcomes are in it?

Step 2

(a) Add the trials

Trials that gave a 6, 7 or 8: 30 + 34 + 30.

Step 3

(a) The experimental probability

94 out of 250, to the nearest thousandth.

Step 4

(b) The theoretical probability

Three winning outcomes out of eight equally likely ones. To the nearest thousandth.

Step 5

(c) The statement

Assuming the machine is fair, which is true?

Q8 Q10

Experimental and theoretical probability

0.376 against 0.375