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

• An outcome is a possible result of an experiment.

• An event is a set of outcomes of an experiment.

• An elementary event is an event which contains only a single outcome in the sample space.

• Consider the experiment of rolling a die. The sample space is \(\{1, 2, 3, 4, 5, 6\}\).

• The event “rolling an even number” is the set \(\{2, 4, 6\}\).

• The event “rolling a number less than 3” is the set \(\{1, 2\}\).

• The event “rolling a number greater than 6” is the empty set \(\emptyset\).

• The probability of an event is a number between 0 and 1 which represents the likelihood of that event occurring.

• The probability of an event \(A\) is denoted by \(P(A)\).

• The probability of the sample space is 1, i.e. \(P(S) = 1\).

• The probability of the empty set is 0, i.e. \(P(\emptyset) = 0\).

• Consider the experiment of rolling a die. The probability of the event “rolling an even number” is \(P(\{2, 4, 6\}) = \frac{1}{2}\).

• The probability of the event “rolling a number less than 3” is \(P(\{1, 2\}) = \frac{1}{3}\).

• The probability of the event “rolling a number greater than 6” is \(P(\emptyset) = 0\).

• For any event \(A\), \(P(A) \geq 0\).

• \(P(S) = 1\).

• The probability of an event not happening is \(1 - P(A)\).

• If \(A_1, A_2, \ldots\) are mutually exclusive events, then \(P(A_1 \cup A_2 \cup \ldots) = P(A_1) + P(A_2) + \ldots\).

• Events \(e_1\) and \(e_2\) are independent if the occurrence of one does not affect the likelihood of the other, defined as \[P(e_1 \cap e_2) = P(e_1)P(e_2).\]

• The notation \(P(e_1 \cap e_2)\) denotes the probability of \(e_1\) and \(e_2\) happening, highlighting the intersection of events.

• The notation \(P(e_1 \cup e_2)\) denotes the probability of either \(e_1\) or \(e_2\) happening, highlighting the union of events.

• Consider flipping a coin twice. The sample space is the following:

Let \(X\) be the number of heads. The probability distribution of \(X\) is the following:

• Consider the time to failure of a light bulb. The sample space is the set of all positive real numbers.

• Discrete: The probability of a particular value \(x\) is \(P(X = x)\). You can just read this directly off the graph or table.

• continuous: The probability of a value falling in a particular range is the area under the curve between the two values.

• The probability of a continuous random variable taking any single value is zero.

• Consider the probability distribution of the number of heads when flipping a coin 10 times.

• What is \(P(X = 7)\)?

• What is \(P(X \leq 3)\)?

• Let \(X\) be th erandom variable that determines movement accuracy in a reaching task.

• \(P(X > 0.6)\)?

• \(P(0.5 < X < 0.6)\)?