• In psychology and cognitive neuroscience, we measure behavior and brain activity.

• Results vary across participants, trials, and conditions – sometimes due to real effects, sometimes due to random chance.

• How do we distinguish meaningful effects from mere chance?

• Probability gives us a framework to describe this variability and assess whether an observed effect is real.

• Suppose we run a cognitive task with two groups.

• One group has an average reaction time of 500 ms, the other 480 ms.

• Is this a real effect or just random variation?

• We need probability to determine if our results are beyond what would be expected by chance alone.

• Suppose a neuropsychologist must assess whether or not a stroke victim has a working memory deficit.

• If the average recall is 7 \(\pm\) 2 items, does this mean everyone recalls between 5 and 9 items?

• No! There’s natural variation — some people recall 4, others 10 etc.

• If the stroke victim recalls 3 items, is this a deficit?

Now that we see why probability is important, let’s define it:

• Sample space: The set of all possible outcomes.

• Outcome: A single result of an experiment.

• Event: A set of outcomes.

• Probability of an event P(A): A number between 0 and 1 representing the likelihood of that event.

The sample space is the complete set of all possible outcomes of an experiment.

• Example (Single Coin Toss): \[ S = \{H, T\} \]

• Example (Two Coin Tosses): \[ S = \{HH, HT, TH, TT\} \]

An outcome is a single, specific result of an experiment.

• Example: Getting “Heads” (H) in a single coin flip.

• Example: Getting “Tails-Heads” (TH) when flipping a coin twice.

An event is a set of outcomes that share some property we care about.

• Example: “Getting at least one Heads” in two flips \[ E = \{HH, HT, TH\} \]

• Example: “Both flips match” (either HH or TT) \[ E = \{HH, TT\} \]

Determining Probability from a Sample Space Table

• The probability of an event can be determined by:
  1. Constructing a table of the full sample space.
  2. Counting the number of rows where the event occurs.
  3. Dividing by the total number of rows.

Assumptions Required

• All outcomes in the sample space are equally likely

• The sample space is complete (i.e., it includes all possible outcomes of the experiment).

• Example: What is the probability of getting at least one Heads?

  • The event “at least one Heads” includes all rows except TTT.

  • There are 7 favorable outcomes out of 8 total.

\[ P(\text{at least one Heads}) = \frac{7}{8} = 0.875 \]

In probability theory, three fundamental axioms define how probabilities behave. These were formalized by Andrey Kolmogorov in 1933.

• Axiom 1: Non-Negativity
• Axiom 2: Normalization
• Axiom 3: Additivity

• The probability of any event \(A\) is always non-negative:

  \[ P(A) \geq 0 \]

• This means probabilities cannot be negative and every event has a probability that is at least zero.

• Example: The probability of getting a “Heads” in a single coin flip is 0.5. Same for “Tails”.

• The probability of the entire sample space \(S\) is always 1:

  \[ P(S) = 1 \]

• This ensures that something must happen — the total probability of all possible outcomes is 1.

• Example: The probability of getting either “Heads” or “Tails” in a single coin flip is 1.

• Example: The probability of getting HH, HT, TH, or TT in two coin flips is \(P(\{HH, HT, TH, TT\}) = 1\).

• If two events \(A\) and \(B\) are mutually exclusive (i.e., they cannot occur together):

  \[ P(A \cup B) = P(A) + P(B) \]

• This states that the probability of either \(A\) or \(B\) happening is simply the sum of their individual probabilities.

• Example: The probability of getting either “Heads” or “Tails” in a single coin flip is \(0.5 + 0.5 = 1\).

• The empty set \(\emptyset\) contains no outcomes, so:

  \[ P(\emptyset) = 0 \]

• Example: The probability of flipping a coin getting neither “Heads” nor “Tails” is 0.

• The probability of not \(A\) (denoted \(A^c\)) is:

  \[ P(A^c) = 1 - P(A) \]

• This follows from the normalization axiom, since every possible outcome must be accounted for.

• Examples: Single Coin Flip

  • Let \(A\) be the event “getting Heads” in a single fair coin flip.

  • Since \(P(A) = 0.5\), the probability of not getting Heads (i.e., getting Tails) is:

    \[ P(A^c) = 1 - P(A) = 1 - 0.5 = 0.5 \]

• Example: Two Coin Flips:

  • Let \(A\) be the event “getting HH” (both flips are Heads).

  • Since \(P(HH) = 0.25\), the probability of not getting “HH” (i.e., getting at least one Tails) is:

    \[ P(A^c) = 1 - P(HH) = 1 - 0.25 = 0.75 \]

  • This accounts for the remaining three outcomes: HT, TH, TT.

• If \(A\) and \(B\) are not mutually exclusive:

  \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

• This corrects for double-counting of \(P(A \cap B)\).

• Example: Coin Flip with Overlapping Events Suppose we flip a fair coin twice, with the sample space:

  \[ S = \{HH, HT, TH, TT\} \]

Let’s define two events:

• \(A\): “At least one heads” \(\rightarrow A = \{HH, HT, TH\}\)

• \(B\): “Heads on the first flip” \(\rightarrow B = \{HH, HT\}\)

Now, calculate probabilities:

• \(P(A) = P(HH) + P(HT) + P(TH) = 0.25 + 0.25 + 0.25 = 0.75\)

• \(P(B) = P(HH) + P(HT) = 0.25 + 0.25 = 0.5\)

• \(P(A \cap B)\) (both events happen, meaning “at least one heads and heads on the first flip”):

• Overlapping outcome: \(P(A \cap B) = P(HH) + P(HT) = 0.25 + 0.25 = 0.5\)

Using Inclusion-Exclusion:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

\[ P(A \cup B) = 0.75 + 0.5 - 0.5 = 0.75 \]

This confirms that “At least one heads” OR “Heads on the first flip” occurs with probability 0.75.

• If two events \(A\) and \(B\) are independent (one does not affect the other):

  \[ P(A \cap B) = P(A) P(B) \]

• Example: The probability of flipping two heads in a row (fair coin):

  \[ P(HH) = P(H) \cdot P(H) = 0.5 \times 0.5 = 0.25 \]