MAT-144 · Mathematical Reasoning Topic 07 · Taxes & Stocks
Topic 07 · Review · Q36

Experimental and theoretical probability for compound events

Compute experimental from the data, theoretical from the math (independent → multiply), then compare. The Law of Large Numbers says the two converge as trials grow.

01 Worked Example Video

Every step of Experimental and theoretical probability for compound events explained on video. Pause anywhere; the embed scrolls independently of the page.

YOUTUBE
02 What This Looks Like on ALEKS

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

3 balls × 3 cards, 70 trials

3 balls (1, 2, 3) and 3 cards (K, Q, J). Ball and card are chosen independently at random.

Find the theoretical P(ball is odd AND card is K or J). Round to the nearest thousandth.

P_theo =
2 coins, 100 trials

Two fair coins are flipped. After 100 trials, HH occurred 28 times.

Find the theoretical P(both heads). Type as a decimal.

P_theo =
Die + spinner

A 6-sided die and a 4-section spinner (A, B, C, D). After 60 trials, “even AND vowel A” occurred 9 times.

Find the theoretical P. Type as a decimal.

P_theo =
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 = favorable trials / total trials from the data table
theoretical = P(A) · P(B) when A and B are independent
Two probabilities for the same event: one from counting actual outcomes (experimental), one from computing the math (theoretical). The Law of Large Numbers says the experimental converges to the theoretical as the number of trials grows. See it live: P(die even AND heads) = 1/4 = 0.25 — watch a thousand trials converge from the noisy 0.27 on the way →
Common slips
(1) Missed compound events in the table. “Odd ball AND K or J” means summing several table cells, not just one. (2) Added instead of multiplied. For independent events, theoretical P = P(A) × P(B), not P(A) + P(B). (3) Expected them to be equal. In 100 trials, the experimental is close to theoretical, not equal to it. They converge as trials → ∞.
04 Practice — Step by Step
Practice this problem step by step
A fair die is rolled and a fair coin is flipped. In 100 trials, the event “die is even AND coin is heads” occurred 27 times. Find experimental and theoretical P, then compare.
Step 1

Favorable count from the data

How many trials (out of 100) had die even AND heads?
favorable =
Step 2

Experimental probability

Compute 27 / 100 as a decimal.
P_exp =
Step 3

Theoretical probability

Compute P(even) · P(heads) = (1/2) · (1/2) as a decimal.
P_theo =
Step 4

Compare — Law of Large Numbers

P_exp = 0.27, P_theo = 0.25. As the number of trials grows very large, what should happen?
Q35 Q37
MAT-144 · TOPIC 07 · REVIEW · Q36 OF 38