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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.
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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?
