Calculating Theoretical Probability of Simple Events - Seventh Grade Mathematics

Theoretical probability calculates the exact mathematical likelihood of an event based on the structure of the sample space and the assumption of equally likely outcomes.

Sample Space and Equally Likely Outcomes

The sample space of an experiment, denoted by the capital letter S, is the complete set of all possible outcomes that can result from that experiment. An outcome is a single possible result of a random trial.

               SAMPLE SPACE NOTATION
   Flipping a fair coin:          S = { Heads, Tails }
   Rolling a standard 6-sided die: S = { 1, 2, 3, 4, 5, 6 }
   Spinning a 4-color spinner:    S = { Red, Blue, Green, Yellow }

Theoretical probability relies on the foundational premise that every basic outcome in the sample space is equally likely to occur. A fair coin is balanced so that heads and tails have identical likelihood. A fair six-sided die has six symmetrical faces of identical area and weight.

The Classical Theoretical Probability Formula

When all outcomes in a finite sample space S are equally likely, the theoretical probability of an event E is the ratio of the number of favorable outcomes to the total number of possible outcomes.

               THE THEORETICAL PROBABILITY FORMULA
                     P(E) = n(E) / n(S)
   n(E) = Number of favorable outcomes satisfying event E
   n(S) = Total number of possible outcomes in sample space S

Consider rolling a standard six-sided die and finding the probability of rolling a prime number. Step 1: Identify the total sample space: S = { 1, 2, 3, 4, 5, 6 }, so n(S) = 6. Step 2: Identify favorable outcomes (prime numbers): 2, 3, and 5 are prime (1 is not prime). The event set is E = { 2, 3, 5 }, so n(E) = 3. Step 3: Calculate the probability: P(prime) = 3 / 6 = 1 / 2 = 0.50 = 50%.

Uniform Probability Models

A probability model consists of a sample space S along with the assigned probabilities for each individual outcome. When all outcomes share the identical probability, it is a uniform probability model.

               STANDARD 52-CARD DECK MODEL
   Total Cards: n(S) = 52
   Suits: Hearts (13), Diamonds (13), Spades (13), Clubs (13)
   Colors: Red (26), Black (26)
   Face Cards: Jacks (4), Queens (4), Kings (4) = 12 total
   Aces: 4

Using this uniform model, we can calculate probabilities instantly:

P(Drawing a Heart)    = 13 / 52 = 1 / 4 = 25%
P(Drawing a King)     = 4 / 52  = 1 / 13
P(Drawing a Red Face) = 6 / 52  = 3 / 26

Non-Uniform Probability Models

When outcomes are not equally likely, the model is non-uniform. In non-uniform models, probabilities are determined by the proportional geometric area or frequency allocated to each outcome.

               A NON-UNIFORM SPINNER
                 +-------------------+
                 | \       Red     / |  Red:    180 deg (1/2)
                 |   \           /   |
                 |     \       /     |
                 | Blue  \   / Yellow|  Blue:    90 deg (1/4)
                 | (90d)   *  (90d)  |  Yellow:  90 deg (1/4)
                 +-------------------+

On this spinner, the total angular rotation is three hundred sixty degrees. Red occupies one hundred eighty degrees, so P(Red) = 180 / 360 = 1/2. Blue occupies ninety degrees, so P(Blue) = 90 / 360 = 1/4. Yellow occupies ninety degrees, so P(Yellow) = 90 / 360 = 1/4. Sum of all probabilities: 1/2 + 1/4 + 1/4 = 1.0.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. A standard six-sided die is rolled once. Calculate the theoretical probability of rolling an odd number strictly greater than two.

Exercise 2. A bag contains twelve red marbles, eight green marbles, and five yellow marbles. One marble is drawn at random. Calculate the probability of drawing a green marble, and the probability of drawing a marble that is not red.

Exercise 3. A letter is chosen at random from the letters in the word MATHEMATICS. Find the probability of choosing the letter M, and the probability of choosing a vowel.

Exercise 4. A circular game spinner is partitioned into eight equal sectors numbered one through eight. Calculate the probability of spinning a multiple of three.

Solutions

Solution to Exercise 1. The sample space is S = { 1, 2, 3, 4, 5, 6 }, with n(S) = 6 possible outcomes. Identify the outcomes that are both odd and strictly greater than two: the odd numbers on a die are 1, 3, and 5; among these, only 3 and 5 are strictly greater than two. The favorable outcomes are { 3, 5 }, so n(E) = 2. Compute the probability: P(E) = 2 / 6, which simplifies to one-third (or approximately 33.3%).

Solution to Exercise 2. Find the total number of marbles in the bag: twelve plus eight plus five equals twenty-five marbles (n(S) = 25). To find the probability of drawing a green marble: there are eight green marbles, so P(green) = 8 / 25 = 0.32 = 32%. To find the probability of drawing a marble that is not red: calculate the marbles that are not red (eight green plus five yellow equals thirteen), or use the complement rule: one minus twelve twenty-fifths equals thirteen twenty-fifths (13/25 = 0.52 = 52%).

Solution to Exercise 3. Count the total letters in MATHEMATICS: M-A-T-H-E-M-A-T-I-C-S has eleven letters, so n(S) = 11. Count the occurrences of the letter M: the letter M appears twice, so P(M) = 2 / 11. Count the vowels (A, E, I): there are two A's, one E, and one I, giving a total of four vowels. The probability of choosing a vowel is four-elevenths (4/11).

Solution to Exercise 4. The spinner has eight equal sectors: S = { 1, 2, 3, 4, 5, 6, 7, 8 }, so n(S) = 8. The multiples of three in this set are 3 and 6, giving two favorable outcomes. Compute the probability: P(multiple of 3) = 2 / 8, which simplifies to one-fourth (1/4 = 0.25 = 25%).