Lecture 2: Introduction to Probability

Probability asks the question “How likely is this to occur?”

Frequently, the study of probability begins through gambling and card games, where it is important to know both “How much can I expect to win?” and “How likely is it that I win?”

Let's look at blackjack. Two cards are dealt, one face up and the other face down.

The act of selecting the cards is called an experiment.

An experiment, as in science, is a set of actions resulting in something that can be measured.

In blackjack, we “measure” the cards that end up in our hand, which is face up and which is face down.

Each time a hand is actually dealt and the cards recorded is a trial.

The distinction is that an experiment is the set-up, or the game plan, and the trial is the text run.

Usually, we will be considering many, many trials.

In any experiment, there are a number of different possible outcomes.

In blackjack, the possible outcomes are the hands that we are given.

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

Since there are lots of possible blackjack hands, let's look at an experiment with fewer outcomes.

I'm going to toss three pennies and record the number of pennies that land with “heads” up.

What is a trial of this experiment? Flipping the coins.

What is the sample space? {no heads, one heads, two heads, three heads}

If I toss a nickel, a dime, and a penny instead of three pennies, what does that change?

The answer, of course, depends on what I now record.

If I still only record the number of heads, then the sample space is the same.

If, instead, I recognize that I have more information available to me and record which coin(s) land with “heads” up, then I now have a larger sample space. {ttt,tth,tht,htt,thh,hth,hht,hhh}

An event is a subset of the sample space.

One event in blackjack, which makes the player very happy, is when the two cards dealt are an ace and either a ten or a face card. There are many ways in which this can happen.

We can also take the event where the cards dealt are the ace of spades and the jack of clubs. There are two ways to do this.

If we are flipping three pennies, we can look at the event where two pennies come up heads.

If we are flipping distinct coins, we can look at the event where any two of them come up heads. This is the event {thh,hth,hht}.

If the outcome of a particular trial is one of the outcomes in the event E, we say that the event E occurs.

What about combining events?

Since events are sets, we can combine them using the set operators we looked at on Wednesday.

The event E union F is the event where the outcome is either in E or in F.

The event E intersect F is the event where the outcome is in both E and F.

Two events are mutually exclusive if their intersection is empty.

The event E' is the event where E does not occur.

Using events, it is easy to see how the two coin tossing experiments relate.

Each penny toss outcome is an event in the distinct coin tossing experiment.

{{ttt},{tth,tht,htt},{thh,hth,hht},{hhh}}

When each outcome in the sample space is equally likely, the probability that a single outcome occurs depends only on the size of the sample space.

The probability of hth is 1/8.

If our sample space is S, the probability is 1/n(S) for any outcome.

Any event has probability n(E)/n(S) of occuring.

3/8 of the time, the outcome is two heads.

How does the probability of two events relate?

In general, p(E union F) = p(E) + p(F) – p(E intersect F)

Example. All heads = E, dime is heads = F

p(E) = 1/8

p(F) = 3/8

p(E union F) = 3/8

p(E intersect F) = 1/8

Now flip five coins. What is the sample space and the probability of each outcome?

Flip them consecutively to make computing the probabilities easier.

The probabilities assigned to outcomes in the sample space are called a probability distribution.

Suppose your sample space is {s_1,s_2, ... , s_n} and the probability of outcome s_i is p_i.

Then 0 <= p_i <= 1 each event can only occur a fraction of the time.

p_1 + p_2 + ... + p_n = 1 there is always an outcome.

We can use Venn diagrams to figure out the probability.

women	.14	.36	.10	long hair	.40 A – women (50%)

B – long hair (46%)

A intersect B – long-haired women (36%)

short-haired men?

Elizabeth Moseman

2005-03-31T13:52:54

Elizabeth Moseman