ASVAB Study Guide
Statistics and Probability · Lesson 3 of 3

Basic Probability

Key rules and formulas, worked examples and 6 practice problems with step-by-step solutions.

1.Theoretical Probability

The likelihood of an event happening based on all possible outcomes.

$$P(E) = \frac{\text{Fav. Outcomes}}{\text{Total Outcomes}}$$

Results are always between 0 (impossible) and 1 (certain).

Example

What is the probability of rolling a 4 on a standard six-sided die?

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Step 1: Identify the number of favorable outcomes: 1 (rolling a 4)

Step 2: Identify the total number of possible outcomes: 6 (numbers 1 to 6)

Step 3: Write as a fraction: $$\frac{1}{6}$$

Practice problems

  1. Problem 1

    A bag contains 3 red marbles, 5 blue marbles, and 2 green marbles. What is the probability of randomly drawing a red marble?

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    Answer: $$\frac{3}{10}$$

    Step 1: Find the total number of marbles: $$3 + 5 + 2 = 10$$ total outcomes

    Step 2: Count the target favorable outcomes (red marbles): 3

    Step 3: Place favorable outcomes over total outcomes: $$\frac{3}{10}$$

  2. Problem 2

    If a card is chosen at random from a standard deck of 52 cards, what is the probability of selecting a heart?

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    Answer: $$\frac{1}{4}$$

    Step 1: Identify total cards in a deck: 52 outcomes

    Step 2: Identify the number of hearts in a standard deck: 13 favorable outcomes

    Step 3: Create the fraction and simplify: $$\frac{13}{52} = \frac{1}{4}$$

  3. Problem 3

    A spinner is divided into 8 equal numbered sections from 1 to 8. What is the probability of landing on an odd number?

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    Answer: $$\frac{1}{2}$$

    Step 1: Count total possible spaces on the spinner: 8 outcomes

    Step 2: Count the odd numbers present $$(1, 3, 5, 7)$$: 4 favorable outcomes

    Step 3: Create the fraction and simplify: $$\frac{4}{8} = \frac{1}{2}$$

Remember

Make sure to verify that the total outcomes count represents all possible events under identical conditions.

2.Complementary Events

The probability of an event NOT occurring.

P(not E) = $$1 - P(E)$$

The sum of an event and its complement is always 1.

Example

If the probability of rain tomorrow is 30% (0.30), what is the probability that it will not rain?

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Step 1: Identify the probability of the primary event: $$P(E) = 0.30$$

Step 2: Set up the subtraction model from the baseline whole: $$1 - 0.30$$

Step 3: Calculate the complementary balance outcome: 0.70 or 70%

Practice problems

  1. Problem 4

    The probability of winning a prize in a contest is 0.05. What is the probability of NOT winning a prize?

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    Answer: 0.95

    Step 1: Identify the given winning probability: $$P(E) = 0.05$$

    Step 2: Apply the formula P(not E) $$= 1 - P(E)$$

    Step 3: Compute the subtraction: $$1 - 0.05 = 0.95$$

  2. Problem 5

    A quality control inspector estimates that 4% of products have defects. What is the probability that a randomly chosen item is NOT defective?

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    Answer: 96%

    Step 1: Convert the percentage to a decimal value: $$4\% = 0.04$$

    Step 2: Subtract the decimal value from 1: $$1 - 0.04 = 0.96$$

    Step 3: Convert the remaining decimal back to a percentage format: $$0.96 \times 100 = 96\%$$

  3. Problem 6

    The probability that a public transit train arrives on time is $$\frac{7}{9}$$. What is the probability that the train will be late?

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    Answer: $$\frac{2}{9}$$

    Step 1: Express the whole baseline value as a fraction: $$1 = \frac{9}{9}$$

    Step 2: Subtract the target fractional rate: $$\frac{9}{9} - \frac{7}{9}$$

    Step 3: Find the difference: $$\frac{2}{9}$$

Remember

Subtract the chance of the event happening from 1 to easily solve the inverse scenario.