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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(event)=number of favorableoutcomestotal numberof outcomesP(\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(not A)=1−P(A)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(blue)=35+3+2=310P(\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(heart)=13/52=1/4P(\text{heart}) = 13/52 = 1/4. By the complement rule:

P(not heart)=1−14=34P(\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(heads on a fair coin)P(\text{heads on a fair coin})

Problem 2

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

Problem 3

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

Problem 4

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

Practice

Standard problems matching the lesson.

Problem 5

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

Problem 6

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

Problem 7

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

Problem 8

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

Problem 9

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

Problem 10

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

Problem 11

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

Problem 12

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

Problem 13

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

Problem 14

100 jellybeans: 25 cherry, 30 lemon, 20 grape, 25 orange. P(grape)\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(rolling a multiple of 3 on a die)P(\text{rolling a multiple of 3 on a die})

Problem 16

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

Problem 17

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

Problem 18

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

Problem 19

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

Problem 20

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

Problem 21

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

Problem 22

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