probability experiment

A probability experiment is an action, or trial, through which specific results (counts, measurements, or responses) are obtained.

Examples:

• Tossing a coin four times.
• Asking 50 randomly selected students in the hallway whether they favor the semester system or trimester system.
• Spinning a spinner whose base is divided into red, blue, yellow, and green sections and then rolling a six-sided die.
  

outcome

An outcome is the result of a single trial in a probability experiment.

Examples

• Tossing a coin four times: {HHTH}
• Spinning a spinner whose base is divided into red, blue, yellow, and green sections and then rolling a six-sided die: {B3}

sample space

The sample space is the set of all possible outcomes of a probability experiment.

Examples

• Tossing a coin four times: {HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT}.
• Spinning a spinner whose base is divided into red, blue, yellow, and green sections and then rolling a six-sided die: {R1, R2, R3, R4, R5, R6, B1, B2, B3, B4, B5, B6, Y1, Y2, Y3, Y4, Y5, Y6, G1, G2, G3, G4, G5, G6}
  

event

An event is a subset of the sample space. It may consiste of one or more outcomes.

Examples

• If you toss a coin four times, one event might be "getting two heads." It would contain these outcomes: {HHTT, HTHT, HTTH, THHT, THTH, TTHH}.
• If you spin a spinner whose base is divided into red, blue, yellow, and green sections and then roll a six-sided die, one event might be getting a yellow on the spinner or a two on the die. It would contain these outcomes: {R2, B2, Y1, Y2, Y3, Y4, Y5, Y6, G2}.

tree diagram

A tree diagram is a visual representation of a probability experiment that aids in construction of a sample space. A table may also be used to aid in construction of sample spaces.

simple event

A simple event is an event that consists of one outcome only.

Classical  probability (or theoretical probability) is used when each outcome in a sample space is equally likely to occur.

P(E) = (# of outcomes in event E)/(total # of outcomes in sample space)

Example

• If you toss a coin four times, P(two heads) = 6/16 = .375. (See cards for "event" and "sample space.")

Empirical probability (sometimes called statistical probability) is based on observations obtained from probability experiments. It is simply the relative frequency of an event.

P(E) = (frequency of event E)/(total frequency)

The complement of event E is the set of all outcomes in a sample space that are not included in event E. We use the symbol E' to stand for the complement of event E, and we say it "E prime."

P(E) + P(E') = 1

Example

If E is the event of tossing three heads when you flip a coin four times, E = {HHHT, HHTH, HTHH, THHH} and E' = {HHHH, HHTT, HTHT, HTTH, HTTT, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT}. See "sample space" card.

fundamental counting principle

The fundamental counting principle states that if one event can happen m ways, and a second event can happen n ways, there are m•n ways for the events to happen in sequence. You can extend this rule to three or more events.

# ways = m•n

Example

• If G1 is number of ways to pick a green sock from drawer 1, and G2 is the number of ways to pick a green sock from drawer 2, then number of ways to pick a green sock from drawer 1 and a green sock from drawer 2 is G1•G2.
  

The probability of an event E is between 0 and 1, inclusive. ("Inclusive" means that it can be 0, and it can be 1.)

0 ≤ P(E) ≤ 1

A conditional probability is the probability of an event occurring given that another event has already occurred.

We use the symbols P (B |A) to represent the conditional probability of event B occurrring given that event A has occurred. We read it as "the probability of B given A."

Two events are independent if the occurrence of one of the events does not affect the probability of the other event occurring.

Two events are indpendent if P (B |A) = P (B ) or if P (A |B) = P (A )

Events that are not independent are dependent.

The probability that two events A and B will occur in sequence is

P (A and B) = P (A )•P (B |A)

If events A and B are independent then the rule can be simplified to

P (A and B) = P (A )•P (B )

This simplified rule can be used for any number of events that occur in sequence. For example: P (A and B and C ) = P (A )•P (B )•P (C )

Two events A and B are mutually exclusive if A and B cannot occur at the same time.

The probability that events A or B will occur is given by

P (A or B) = P (A ) + P (B ) - P (A and B)

If events A and B are mutually exclusive then the rule can be simplified to

P (A or B) = P (A ) + P (B )

This simplified rule can be used for any number of events that occur in sequence. For example: P (A or B or C ) = P (A ) + P (B ) + P (C )