Probability is simply the likelihood that an event will occur given certain parameters.

• Experiment – is the action or process with an observable outcome.

  • Examples: flipping a coin, rolling dice, removing a colored ball from a box, etc.

• Sample space(S) – The set of all possible outcomes of an experiment.

  • Example: Head or Tails, 1, 2, 3, 4, 5, 6 for the roll of a die, removing the blue or red ball from the box.

• Event(E) – an instance of the sample space.

  • Example: flipping the coin once,rolling the dice once and receivinge a given number, or removing the blue ball from the box.

• Probability of an Event P(E) – The likelihood that a specified event occurs.

• P(E)=Number of desired outcomes/Total number of possible outcomes

  • Example: Flipping a coin and getting heads

    • P(heads)=\(1/2\)=0.5

    • Number of desired outcomes = 1. The only outcome we are looking for is heads.

    • Total number of possible outcomes = 2. The options are heads and tails.

• Theoretical Probability: Based on reasoning or calculation

  • Example: Roll an even number on a dice. \(\rightarrow\) P(even) = 3/6 = 0.5

• Experimental Probability: Based on actual experiments.

  • Example: Roll a dice 10 times, get the number 6 three times \(\rightarrow\) P(6)=3/10

• Subjective Probability: Based on belief or experience.

  • Example: I think there is a 70% our football team will win tonight’s game.

• Rule 1: \(0 \le P(E) \le 1\)
  • Probability can neither be negative or greater than 1.
• Rule 2: Sum of probabilities of all outcomes =1.

  • Example: Coin \(\rightarrow\) \(1/2+1/2=1\)

  • Example: Dice \(\rightarrow\) \(1/6+1/6+1/6+1/6+1/6+1/6=1\)

• Complement Rule: \(P(E')=1-P(E)\)

  • The probability that a desired event does not happen.

  • Example: Probability of not rolling a 1 or 2 \(\rightarrow\) \(1-1/6-1/6=4/6\)

• There are a number of ways to visualize probability.

• To compare probabilities of discrete events, use a bar or pie chart.

• To show step-by-step possibilities for multiple events, use a tree-diagram.

• To show overlapping events (union/intersection), use a Venn diagram.

• What is the probability of rolling a 6 when rolling a die?

  library(ggplot2)
  library(formatR)
  
  df <- data.frame( 
    roll = 1:6, probability = rep(1/6,6)
  )
  ggplot(df, aes(x=factor(roll), y=probability, 
                 fill = roll ==6)) +
    geom_bar(stat = "identity") +
    scale_fill_manual(
      values = c("#48D1CC","#98FF98")) +
    labs(x = "Rolled Value",  y = "Probability", 
         title = "Probability of Rolling a 6") +
    theme_get()

• What is the probability of rolling a dice 100 times? How many times did you roll a 6?

  [1] 18

  [1] 0.18

  

If you flip a coin 150 times how many times will you get the observation of Tails?