Sample spaces and events

Outcomes, events and probabilities

A random experiment has a sample space $\Omega$ , the set of all possible outcomes: the six faces of a die, or whether a stock closes up, down or flat. An event is a set of outcomes you care about, such as “”. A probability $P$ assigns each event a number between 0 and 1, with $P(\Omega)=1$ , and probabilities of events that cannot happen together add up.

P(A\cup B)=P(A)+P(B)-P(A\cap B),\qquad P(A^{c})=1-P(A)

(1)

Two events are mutually exclusive when $A\cap B$ is empty, and independent when $P(A\cap B)=P(A)P(B)$ : knowing one tells you nothing about the other. These are different ideas: mutually exclusive events with positive probability are strongly dependent, because one happening rules the other out. For equally likely outcomes, $P(A)$ is the number of outcomes in $A$ divided by the total, so counting is the tool.

Counting and the complementTwo dice sum to 7 in 6 of 36 equally likely ways, so the probability is

6/36=1/6

. Three independent trades each win with probability 0.6. “” is awkward to add up, but its complement “” is a single product:

0.4^3=0.064

, so

P(\text{at least one})=1-0.064=0.936

Sample space: every possible outcome. Event: a set of outcomes.
Mutually exclusive: cannot happen together. Independent: $P(A\cap B)=P(A)P(B)$ .
Complement: $P(A^{c})=1-P(A)$ ; use it for “”.

Exclusive does not mean independent“” and “” are mutually exclusive, so knowing it closed up tells you for sure it did not close down: they are as dependent as events can be. Independence needs the product rule, and is what you assume (often wrongly) when you multiply probabilities of losses across positions.

Lab · what's graded

Counting outcomes15 pts
The addition rule15 pts
Mutually exclusive events15 pts
Complement and independence15 pts
The vocabulary15 pts
At least one winner15 pts
Why use the complement10 pts

Outcomes, events and probabilities

P(A\cup B)=P(A)+P(B)-P(A\cap B),\qquad P(A^{c})=1-P(A)

(1)

Counting and the complementTwo dice sum to 7 in 6 of 36 equally likely ways, so the probability is

6/36=1/6

. Three independent trades each win with probability 0.6. “” is awkward to add up, but its complement “” is a single product:

0.4^3=0.064

, so

P(\text{at least one})=1-0.064=0.936

Sample space: every possible outcome. Event: a set of outcomes.
Mutually exclusive: cannot happen together. Independent: $P(A\cap B)=P(A)P(B)$ .
Complement: $P(A^{c})=1-P(A)$ ; use it for “”.

Lab · what's graded

Counting outcomes15 pts
The addition rule15 pts
Mutually exclusive events15 pts
Complement and independence15 pts
The vocabulary15 pts
At least one winner15 pts
Why use the complement10 pts

Outcomes, events and probabilities

Lab · what's graded

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Sample spaces and events

Outcomes, events and probabilities

Lab · what's graded

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