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Subsection 3.2.2 Probability Spaces

Learner Experience 3.2.12.

Work in groups
Materials needed: \(2\) coins

(a)

Toss two coins simultaneously and record the result of each trial. Repeat the experiment at least \(15\) times.

(b)

Write down all the different possible outcomes you can get. If the first coin shows Heads and the second coin shows Tails, write this outcome as HT.

(c)

Do you think there could be any other possible outcomes that you haven’t seen yet even if they haven’t appeared in your tosses so far?

(d)

Have you already listed every single possible combination that could ever happen when tossing two coins? How do you know your list is complete?

(e)

List the outcomes that match each of the following events:
  1. Both coins show heads.
  2. Exactly one coin shows heads.
  3. Neither coin shows heads (both show tails).

Key Takeaway 3.2.13.

A random experiment is a process that can be repeated and has a well-defined set of possible outcomes, but the outcome of any particular trial cannot be predicted with certainty.
Examples of random experiments include tossing a coin, rolling a die, or drawing a card from a deck.
Sample space is a set containing all possible outcomes of a random experiment. It is usually denoted by \(\Omega\) or S
Outcome is a single result from a trial of a random experiment.
Event is a collection of outcomes from the sample space. An event can consist of one outcome or multiple outcomes. For example, if we are tossing a coin, the event "getting heads" consists of the single outcome H, while the event "getting at least one head when tossing two coins" consists of the outcomes HH, HT, and TH.
The probability of any outcome is a number between \(0\) and \(1\text{.}\) The probabilities of all the outcomes add up to \(1\text{.}\)
The probability of any event \(A\) is the sum of the probabilities of the outcomes in \(A\text{.}\)

Example 3.2.14.

Consider the experiment of rolling a 6-sided die.
  1. What is the sample space for this experiment?
  2. Write down the event of observing an even number.
  3. Calculate the probability of observing an even number.
Answer.
  1. The sample space is \(S = \{1, 2, 3, 4, 5, 6\}\text{.}\)
  2. The event of observing an even number is \(A = \{2, 4, 6\}\text{.}\)
  3. \(\displaystyle P(A) = \frac{|A|}{|\Omega|} = \frac{3}{6} = \frac{1}{2} = 0.5 = 50\%\)

Example 3.2.15.

Consider the experiment of rolling a pair of \(6\)-sided dice. For each die, we can observe a number from \(1\) to \(6\text{.}\) If we paired the observations from each die, we would have a single observation from the pair. For example, if the first die lands on \(2\) and the second lands on \(5\text{,}\) we can write down this outcome as (2,5).
  1. What is the sample space for this experiment?
  2. Write down the event of observing the same number on both dice.
  3. Write down the event of observing numbers that sum to \(4\text{.}\)
Answer.
  1. \(S = \{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), \) \((2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), \) \((4,2), (4,3),(4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), \) \((5,6), (6,1), (6,2),(6,3), (6,4), (6,5), (6,6)\}\)
  2. \(\displaystyle A = \{ (1,1),(2,2), (3,3), (4,4),(5,5),(6,6) \}\)
  3. \(\displaystyle B = \{ (1,3), (2,2) , (3,1)\}\)

Checkpoint 3.2.16.

Checkpoint 3.2.17.

Exercises Exercises

1.

A standard six-sided die is rolled once.
  1. List the complete sample space \(S\) for this experiment.
  2. List the outcomes that make up the event \(E\text{:}\) "Rolling a prime number."
  3. List the outcomes for the event \(F\text{:}\) "Rolling a number greater than 4."
Answer.
  1. \(\displaystyle S = \{1, 2, 3, 4, 5, 6\}\)
  2. \(\displaystyle E = \{2, 3, 5\}\)
  3. \(\displaystyle F = \{5, 6\}\)

2.

A bag contains three numbered tiles: \(1, 2,\) and \(3\text{.}\) Two tiles are drawn one after the other with replacement (the first tile is put back before the second is drawn).
  1. Write out the sample space as a set of ordered pairs.
  2. Identify the outcomes in the event: "The sum of the two tiles is exactly 4."
Answer.
  1. \(\displaystyle S = \{(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)\}\)
  2. Event outcomes: \(\{(1,3), (2,2), (3,1)\}\)

3.

A small cafeteria offers a "Lunch Combo" where a student chooses one main dish from {Burger, Pizza} and one drink from {Juice, Water, Milk}.
  1. List the sample space \(S\) of all possible lunch combinations.
  2. Identify the outcomes in the event \(A\text{:}\) "The lunch includes either Pizza or Milk (or both)."
Answer.
  1. \(\displaystyle S = \{(B, J), (B, W), (B, M), (P, J), (P, W), (P, M)\}\)
  2. Event outcomes: \(\{(B, M), (P, J), (P, W), (P, M)\}\)

4.

Three children are born in a family. Let \(B\) represent a boy and \(G\) represent a girl.
  1. List the sample space for the gender of the three children in order of birth.
  2. List the outcomes for the event \(E\text{:}\) "There are at least two girls."
  3. Identify the outcomes for the event \(F\text{:}\) "The firstborn is a boy."
Answer.
  1. \(\displaystyle S = \{BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG\}\)
  2. Event \(E\text{:}\) \(\{BGG, GBG, GGB, GGG\}\)
  3. Event \(F\text{:}\) \(\{BBB, BBG, BGB, BGG\}\)

5.

In an experiment, a spinner with three colors (Red, Blue, Green) is spun, and then a coin is tossed.
  1. List the sample space using notation like \(RH\) for "Red and Heads."
  2. Identify the event where the spinner lands on a primary color (Red or Blue) and the coin lands on Tails.
Answer.
  1. \(\displaystyle S = \{RH, RT, BH, BT, GH, GT\}\)
  2. Event outcomes: \(\{RT, BT\}\)
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