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Sample Space Examples and The Counting Principle

Probability and Statistics > Probability > Sample Space Examples

What is a Sample Space?

The sample space of an experiment is all the possible outcomes for that experiment. A couple of simple examples:

graphics-coins-691670 The space for the toss of one coin: {Heads, tails.}
graphics-dice-231882 The space for the toss of a die: {1, 2, 3, 4, 5, 6.}
graphics-cards-778788 The space for choosing a card from a standard deck:
{Hearts: 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k, A.
Clubs: 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k, A.
Spades: 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k, A.
Diamonds: 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k, A.}
sample space The sample space for the roll of TWO dice:
{[1][1], [1][2], [1][3], [1][4], [1][5], [1][6],
[2][1], [2][2], [2][3], [2][4],[2][5], [2][6],
[3][1], [3][2], [3][3], [3][4], [3][5], [3][6],
[4][1], [4][2], [4][3], [4][4], [4][5], [4][6],
[5][1], [5][2], [5][3], [5][4], [5][5], [5][6],
[6][1], [6][2], [6][3], [6][4], [6][5], [6][6].}

When writing out a sample space, remember this important fact: each item in the sample space should be equally likely. In the sample space above, each of the 36 dice rolls is equally likely to happen. You could write the sample space another way, by just adding up the two dice. For example [1][1] = 2 and [1][2] = 3. That would give you a sample space of {2, 3, 4, 6, 7, 8, 9, 10, 11, 12}. But the problem is, these events aren’t equally likely. You have a 1/36 chance of rolling a 12 (6,6) and a greater chance (4/36) of rolling a seven (4,3 or 3,4 or 5,2 or 2,5).

Large Sample Spaces.

The more items you throw in the mix, the more complicated it becomes. How do you make sure you don’t miss an item in the sample space?

Notice how the sample space for the roll of two dice starts with the number “1” first:
[1][1], [1][2], [1][3], [1][4], [1][5], [1][6]
All of the possible combinations for the second die are listed after the number 1.
Then comes “2”:
[2][1], [2][2], [2][3], [2][4],[2][5], [2][6]
And so on. The problem is, sample spaces can get very, very large. Imagine trying to figure out the sample space for rolling six dice. There are 46656 items in the sample space! There is no fast way to make a sample space — you just have to write out all of the possibilities. However, there is a way you can figure out probabilities of choosing an item from a sample space. Rather than writing out the entire sample space, you can use the Counting Principle.

The Counting Principle.

Sample problem: If shoes come in 6 styles with 3 possible colors, how many varieties of shoes are there?

All you need to do is multiply: 6 • 3 = 18 possible varieties of shoes.

The Fundamental Counting Principle: If there are “a” ways for one event to happen, and “b” ways for a second event to happen, then there are “a * b” ways for both events to happen.

Examples:

  1. Flipping a coin and rolling a die:
    There are 2 ways to flip a coin and 6 ways to roll a die, so there are 6 * 2 = 12 ways to flip a coin and roll a die.
  2. Drawing three cards from a standard deck without replacing the cards.
    There are 52 ways to draw the first card.
    There are 51 ways to draw the second card.
    There are 50 ways to draw the third card.
    There are 52 * 51 * 50 = 132,600 ways to draw the three cards.

You can see why you might not want to draw out the entire sample space. Can you imagine writing out all 132,600 possibilities?

The Counting Principle also works for more than two events.

  1. Rolling a die four times:
    There are 6 ways to roll each die.
    There are 6 * 6 * 6 * 6 = 1,296 possible outcomes.
  2. Taking a 10 question survey with 2 choices (yes or no) for each question:
    There are 2 ways to answer each question.
    There are 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 1024 ways to complete the survey.
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Sample Space Examples and The Counting Principle was last modified: October 12th, 2017 by Stephanie Glen

2 thoughts on “Sample Space Examples and The Counting Principle

  1. Kolawole Mayowa Junior

    It really helped a lot, i’ve been trying to differentiate & similarize between sample space and probability but with this, i know i got already. Thnks for your help