4.2 Probability Models

We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

Experiments Consider tossing a fair coin 5 times. Let H represent obtaining a head and T represent obtaining a tail. Then one possible outcome is HHTHT. This is an example of an experiment; that is, any activity that yields a result or an outcome. A survey with yes/no/undecided outcomes is an experiment.

Sample Space If we consider the coin tossing experiment mentioned above, the possible outcomes are HHHHH, HHHHT, HHHTH, HHTHH, HTHHH, THHHH, HHHTT, HHTHT, HTHHT, etc. The collection of all possible distinct outcomes that can occur when an experiment is performed is called the sample space. The sample space must have the property that when the experiment is performed, exactly one of these outcomes must occur. We may wish to give the sample space a name (say S) and we usually write the sample space inside of brackets. In the case of a single toss of a coin, the sample space would be written S={H,T}.

Events An event is a subset of the sample space. An event can be a single outcome or several. In the case where the event is a single outcome, we call this a simple event. If we let S={H,T}, then an event can be H or TTTTH. If we let S={HHHHH, HHHHT,HHHTH,HHTHH, …} then the event TTTTH is simple.

What is probability? Suppose that A is some event. The probability of A, denoted P(A), is the expected proportion of occurrences of A if the experiment were to be repeated many times. In other words, if S is the sample space and A is an event then

Suppose we roll a fair die 2 times. What is the probability that the product of the two rolls is divisible by 3? This requires finding not only the size of the sample space, but also the number of outcomes that satisfy our condition. We’ll use brute force. S={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2, 3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6 ),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

Listing out all possibilities and then counting one by one is admittedly not a very good way of doing things. Much of what we discuss in the sections on probability will therefore be concerned with convenient was to count. How many distinct strings comprised of three letters can be made out of the 26 letters of the alphabet?

Multiplication Rule This leads us to the Fundamental Principle of Counting or the Multiplication Rule: If Task 1 can be performed in n ways and Task 2 can be performed in m ways, then Task 1 and Task 2 can be performed together in nm ways. How many ways are there to toss a fair die 5 times with each roll showing a different number than the previous rolls?

We have looked at P(A); that is, the probability of a single event A taking place e.g. when rolling a die 5 times, what is the probability that the sum of the 5 rolls will be divisible by 4? Call this event A. Now let B be the event of rolling a die 5 times and obtaining an even number on every roll. What is the probability of both A and B happening? Or what is the probability that at least one of A or B happens? Or what is the probability that A does NOT happen? These are compound events.

Language, Truth, and Logic Let A and B be events from a sample space S. A or B is the event that either A occurs or B occurs or both. This event is usually denoted AυB, read A union B or A or B. A and B is the event that both A occurs and B occurs at the same time. This event is usually denoted A∩B, read A intersect B or A and B. The complement of A is the event that an outcome in S that is NOT in A will occur. The event is denoted Ā, read the complement of A. We may illustrate these with Venn diagrams.

Disjoint Events Consider a deck of 52 playing cards (13 spades, 13 clubs, 13 hearts, 13 diamonds) and consider an experiment where we draw one card at random. Let E be the event of drawing a black card with an even value (2,4,6,8,10) and D be the event of drawing a diamond. What is the probability of E υ D (E or D)? Events A and B are disjoint events if they can not occur together when the experiment is performed.

The Addition Rule Suppose A and B are disjoint events. Then P(A or B)=P(A)+P(B). If C is a third event which is mutually exclusive with both A and B, then P(A or B or C)= P(A)+P(B)+P(C) etc. If a pair of dice is rolled, find the probability of rolling a double or getting a sum of 9.

Independent events Consider an experiment where a card is randomly drawn from a deck (52 cards), recorded, replaced, the deck is shuffled, and another card is drawn. Let A be the event of drawing a face card (king, queen, jack) on the first draw and B be the event of drawing a red card on the second draw. What is the probability of A and B?

Events A and B are said to be independent if the occurrence of one has no effect on the probability of the occurrence of the other. Suppose A and B are independent events. Then P(A and B)=P(A)P(B). Suppose two fair dice are rolled. What is the probability that the first one will show an even number and the second one will show an odd number? The Monty Hall Problem

Complementary Events Let A be an event. Then Ā is any event in S which is not in A. Hence P(A)+P(Ā)=1 and so P(Ā)=1-P(A). This fact can often make problems easier. Suppose we roll three 20-sided dice (we’re playing D & D). What is the probability that no single die will show a 20?