Statistics

Probability Fundamentals

Lesson

Probability measures how likely something is. It runs from 0 (impossible) to 1 (certain). When all outcomes of a random experiment are equally likely, computing a probability is a counting problem:

P(\text{event}) = \frac{\substack{\text{number of favorable}\\\text{outcomes}}}{\substack{\text{total number}\\\text{of outcomes}}}

The sample space is the list of all possible outcomes. An eventis some subset of that — the outcomes you’re asking about.

A useful shortcut — the complement rule:

P(\text{not } A) = 1 - P(A)

Sometimes counting the “not” outcomes is easier than counting the ones you want directly.

Worked example 1

A bag holds 5 red, 3 blue, and 2 green marbles. Draw one at random. What’s the probability it’s blue?

P(\text{blue}) = \frac{3}{5 + 3 + 2} = \frac{3}{10}

Worked example 2 — complement

Draw a card from a standard deck. What’s the probability it’s not a heart?

Hearts are 13 of 52 cards, so $P(\text{heart}) = 13/52 = 1/4$ . By the complement rule:

P(\text{not heart}) = 1 - \tfrac{1}{4} = \tfrac{3}{4}

How to type your answer

Type a fraction in lowest terms or the decimal — either works. Examples: 1/2, 0.5, 1/6, 0.7.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

P(\text{heads on a fair coin})

Problem 2

P(\text{rolling a 4 on a fair die})

Problem 3

\text{Bag of 4 red and 6 blue marbles. } P(\text{red})

Problem 4

\text{Standard deck. } P(\text{heart})

Practice

Standard problems matching the lesson.

Problem 5

P(\text{rolling an even number on a die})

Problem 6

P(\text{rolling a number greater than 4 on a die})

Problem 7

\text{Spinner: 3 red, 2 blue, 2 green, 1 yellow (8 sections). } P(\text{red})

Problem 8

\text{Same spinner. } P(\text{blue or green})

Problem 9

\text{Standard deck. } P(\text{king})

Problem 10

\text{Standard deck. } P(\text{face card, J/Q/K})

Problem 11

\text{5 red, 3 blue, 2 green marbles. } P(\text{blue})

Problem 12

\text{Same bag. } P(\text{not red})

Problem 13

\text{Class: 12 boys, 18 girls. } P(\text{girl})

Problem 14

\text{100 jellybeans: 25 cherry, 30 lemon, 20 grape, 25 orange. } P(\text{grape})

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

P(\text{rolling a multiple of 3 on a die})

Problem 16

P(\text{sum of 7 on two fair dice})

Problem 17

P(\text{sum of 11 on two fair dice})

Problem 18

\text{Forecast says 30\% chance of rain. } P(\text{no rain})

Problem 19

\text{Standard deck. } P(\text{not a heart})

Problem 20

\text{Bag: 7 red, 8 blue, 5 green. } P(\text{red or green})

Problem 21

\text{Survey: 60 like pizza, 40 don't. } P(\text{doesn't like pizza})

Problem 22

P(\text{both heads on two coin flips})

Practice

Standard problems matching the lesson.

Problem 23

Draw one card from a standard 52-card deck. P(the card is a heart)?

Problem 24

A bag holds 5 red, 3 blue, and 2 green marbles. Pick one at random. P(blue)?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

Roll two fair six-sided dice. P(the sum is 7)?

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