What is the sample space if the experiment consist s of measuring (in hours) the life time of a transistor?

S=

The sample space consist of all nonnegative real numbers. (This is a continuous sample space)

An event is any collection (subset) of outcomes contained in the sample space S.

• Simple if it consists of exactly one outcome (point).
• Compound it consists of more than one outcome.

Probability Notation

• We use the notation P(event) = p to denote the probability of an event occurring and (1-p) it will not occur
• Probability of heads occurring is P(heads)
• P(heads) = 0.4 indicates an unfair coin that turns up head 40% of the time.

The odds in favour of an event A is defined by \(\frac{P(A)}{P(A^c)} = \frac{P(A)}{1- P(A)}\)

Example
If P(A) = 2 / 3 what are the odds of A (odds in favour of A)?
1 - P(A) =1/3 Odds (A) = \(\frac{P(A)}{1-P(A)}\) = 2 : 1

Let S = { \(a_1, a_2,...a_n\)} be a sample space.
Let P (\(a_i\)), i= 1, 2,…, n be the probabilities to the \(a_i\)’s.

The probability of an arbitrary compound event A can be determined by summing the probabilities of simple events in A.
A = { \(a_1, a2,...a_k\)}

If each simple event has probability \(\frac{1}{n}\) ( i.e. “equally likely”).
P(A) = \(\frac{k}{n}\)

Example
Suppose a 6-sided fair die is rolled, what is the probability of getting an even number ?
Let A = “even number” A = {2, 4, 6}
P (A) = P (2) + P (4) + P (6) = 1/2.

Rules of Probability

Rule 1

\(\sum\) (Of all i) P (\(a_i\)) =1 , P (S) = 1

Rule 2

For any event A, 0 ≤ P(A) ≤ 1
Probabilities are always between 0 and 1
0: event never happens,
1: event always happens.

Rule 3

If A and B are two events with A ⊆ B
(that is, all of the points in A are also in B)
then P(A) ≤ P(B)

Mutually Exclusive or Disjoint Events

Two events are mutually exclusive

• If they cannot happen simultaneously.
• If they cannot occur at the same time.

Example
Tossing a coin once, which can result in either heads or tails, but not both.