Probability Notes

Defining Probability

Sample Space

• The sample space is a list of all possible outcomes of an experiment.

  • It can be small or large depending on the experiment.

• Examples:

  • Flipping a coin: The sample space is heads or tails.
  • Rolling a die: The sample space is 1, 2, 3, 4, 5, or 6.

Social Security Number Example

• A social security number has the format: XXX-XX-XXXX.
• Each digit can be one of 10 possibilities (0-9).
• To find the total number of possible social security numbers, multiply the possibilities for each digit together.
  • $10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10^9 = 1,000,000,000$ (1 billion)
  • There are a billion different possible social security numbers.
• In this case, the sample space would be a list of all those billion possible social security numbers.

Events

• An event is a specific outcome or set of outcomes within the sample space.
• Visually:
  • The sample space is represented as a large rectangle.
  • An event is a smaller part of that rectangle.

Probability Calculation

• Probability is calculated by dividing the number of ways the event can happen by the number of outcomes in the sample space.

  • $Probability = \frac{Number \ of \ ways \ the \ event \ can \ happen}{Number \ in \ the \ sample \ space}$

• Example:

  • Rolling a die and wanting an even number.
    • Sample space: 1, 2, 3, 4, 5, 6 (6 total outcomes).
    • Event: rolling an even number (2, 4, or 6 - 3 outcomes).
    • $Probability = \frac{3}{6} = \frac{1}{2} = 0.5 = 50\%$. The probability of rolling an even number is 50%.

Properties of Probability

• Probabilities are always fractions or numbers between 0 and 1 (inclusive).
  • $0 \le Probability \le 1$
• An impossible event has a probability of 0.
  • Example: rolling a 7 on a standard 6-sided die.
• A certain event has a probability of 1.
  • Example: rolling a number less than 7 on a standard 6-sided die.

Visual Representation of Probability

• Imagine throwing darts at the sample space rectangle.
• A likely event occupies a large portion of the sample space.
• An unlikely event occupies a small portion of the sample space.

Example: Bag of Marbles

• A bag contains 100 marbles.
  • 10 are red.
  • 20 are blue.
  • 30 are green.
  • 40 are of another color.
• The experiment is drawing one marble from the bag.

Visual Representation

• Draw circles representing the number of each color of marble inside the sample space rectangle, showing relative portions.

Calculating Probabilities

• Probability of drawing a red marble:
  • $P(Red) = \frac{10}{100} = \frac{1}{10} = 0.1 = 10\%$
• Probability of drawing a red or green marble:
  • The keyword "or" means either red or green satisfies the event.
  • $P(Red \ or \ Green) = \frac{10 + 30}{100} = \frac{40}{100} = 0.4 = 40\%$.
• Probability of drawing a marble that is not blue:
  • $P(Not \ Blue) = \frac{100 - 20}{100} = \frac{80}{100} = 0.8 = 80\%$.
• Probability of drawing a red or not green marble:
  • $P(Red \ or \ Not \ Green)$
  • There are 10 red marbles.
  • There are 70 marbles that are not green (100 total - 30 green = 70).
  • If you simply add 10 + 70 = 80, you’re double-counting the red marbles because they are already part of the “not green” group.
  • Therefore, the correct calculation is:
    • $P(Red \ or \ Not \ Green) = \frac{70}{100} = 0.7 = 70\%$.

Key Takeaway

• Probability is about determining the size of the event relative to the size of the sample space.
• $Probability = \frac{Event \ size}{Sample \ Space \ size}$
• Careful consideration is needed to avoid double-counting when calculating the size of the event.