RANDOMNESS Random is not the same as haphazard or helter-skelter or

higgledy-piggledy.

Random events are unpredictable in the short-term, but lawful and well behaved in the long-run.

• For example, if I toss one coin, I do not know whether it will land heads or tails. But if I toss a million coins, I can be reasonably certain that about half of them will be heads and the other half tails.

PROBABILITY Probabilities are numbers which describe the

outcomes of random events.

The probability of an event is the long-run relative frequency of that event.

P(A) means “the probability of event A.”

If A is certain, then P(A) = one

If A is impossible, then P(A) = zero

Sample Space A “sample space” is a list of all possible outcomes

of a random process. – When I roll a die, the sample space is {1, 2, 3, 4, 5, 6}.

– When I toss a coin, the sample space is {head, tail}.

An “event” is one or more members of the sample space. – For example, “head” is a possible event when I toss a

coin. Or “number less than four” is a possible event when I roll a die.

Probability Rules All probabilities are between zero and one:

• 0 < P(A) < 1

Something has to happen:• P(Sample space) = 1

The probability that something happens is one minus the probability that it doesn’t:

• P(A) = 1 - P(not A)

Examples The probability that I wear a green shirt tomorrow is some

number between zero and one.

• 0 < P(green shirt) < 1 The probability that I wear a shirt of some color tomorrow

is equal to one.

• P(shirt) = 1 The probability that I wear a green shirt tomorrow is one

minus the probability that I don’t wear one.

• P(green shirt) = 1 - P(non-green shirt)

CHANCES and ODDS Chances are probabilities expressed as percents. Chances

range from 0% to 100%.

• For example, a probability of .75 is the same as a 75% chance.

The odds for an event is the probability that the event happens, divided by the probability that the event doesn’t happen. Odds can be any positive number.

• For example, a probability of .75 is the same as 3-to-1 odds.

Independence Events A and B are independent if the probability of event B is

not affected by A’s occurring or not occurring:

• For example, if I am tossing two coins, the probability that the second coin lands heads is always .50, whether or not the first coin lands heads.

• P(H2 after H1) = P(H2 after T1) = P(H2)

The Addition Rule

If A and B cannot both occur, then

• P(A or B) = P(A) + P(B)

• P(green shirt or blue shirt) = P(green shirt) + P(blue shirt)

• The events “green shirt” and “blue shirt” are called disjoint.

The Multiplication Rule

If A and B are independent, then

• P(A and B) = P(A) x P(B)

• For example, if I choose my shirts and pants separately, then:

• P(green shirt and blue pants) = P(green shirt) x P(blue pants)

THE ADDITION RULE for more than two disjoint events

If A and B and C are mutually disjoint, then

• P(A or B or C) = P(A) + P(B) + P(C)

• P(green or blue or white shirt) • = P(green shirt) + P(blue shirt) +

P(white shirt)

THE MULTIPLICATION RULE for more than two independent events

If A and B and C are mutually independent, then

• P(A and B and C) = P(A) x P(B) x P(C)

• If I pick shirts, pants, and belts independently:

• P(green shirt and blue pants and black belt)

• = P(green shirt) x P(blue pants) x P(black belt)