Week 2 Section 1.2-1.4 Texas A& M University Department of Mathematics Texas A& M University, College Station 22 January-24 January 2019 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 1 / 19
Section 1.2 The Number of Elements in a Finite Set In this section, we understand the relationship between the number of elements in A B and the number of elements in A, B and A B. Notation: If A is a set with a finite number of elements, we denote the number of elements in A by n(a). s: Venn Diagram: Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 2 / 19
Union Rule for the Sets Remark If A and B are finite sets, then n(a B) = n(a) + n(b) n(a B) Moreover, if A and B are disjoint, then n(a B) = n(a) + n(b) If n(a) = 100, n(b) = 75 and n(a B) = 40, what is n(a B)? Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 3 / 19
In a survey of 294 people, a pet food manufacturer found that 40 owned a dog but not a cat, 117 owned a cat but not a dog, and 44 owned neither a dog or a cat. a) How many owned both a cat and a dog? b) How many owned a dog? Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 4 / 19
Counting with Three Sets A survey of 1000 adults found that in the past month 500 had been to Burger King, 700 to McDonald s, 400 to Wendy s, 300 to Burger King and McDonald s, 250 to McDonald s and Wendy s, 220 to Burger King and Wendy s and 100 to all three. a) How many went to Wendy s but not the other two? b) How many went to only one of them? b) How many went to none of these three? Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 5 / 19
Counting with Three Sets If n(b) = 200 and n(a B C) = 40, n(a B C c ) = 20, n(a c B C) = 50, what is n(a c B C c )? Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 6 / 19
Section 1.3-Sample Spaces and Events Our aim is to understand the basics of the probability. We relate the concept of the probability to the set theory. An experiment is an activity that has observable results. s: An outcome is the result of the experiment. s: Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 7 / 19
A sample space of an experiment is the set of all possible outcomes of the experiment. s: Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 8 / 19
Tree Diagrams In the last exercise, we completed a task (flipped coin) and then completed another task (flipped coin again). In these cases, the experiment can be diagrammed with a tree. Draw the tree diagram of the following event and show the sample space: A fair die is rolled. If the die show 1 or 6, a coin is tossed. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 9 / 19
Events Given a sample space S for an experiment, an event is any subset E of S. An elementary or simple event is an event with single outcome. A fair coin is flipped twice to observe whether heads or tails falls. (a) What is the sample space S? (b) Find the event that at least one head comes up. (b) Find the event that exactly two tails come up. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 10 / 19
We can use our set language for union, intersection and complement to describe events. Union of Two Events: If E and F are two events, then E F is the union of two events and consists of the set of outcomes that are in E or F. Remark E F occurs means either E or F occurs. Intersection of Two Events: If E and F are two events, then E F is the intersection of two events and consists of the set of outcomes that are in both E and F. Remark E F occurs means both E or F occur. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 11 / 19
If E is an event, then E c is the complement of E and consists of the outcomes that are not in E. Remark The event E c occurs means E does not occur. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 12 / 19
Two dice identical except that one is green and the other is red are tossed and the number of dots on the top face of each is observed. Let E consist of these outcomes for which the number of dots on top faces of both dice is 2 or 4. Let F be the event that the sum of the number of dots on the top faces of the two dice is 6. Let G be the event that the sum of the number of dots on the top faces of the two dice is less than 11. (a) List the elements of E and F. (b) Find E F. (c) Find E F. (d) Find G c. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 13 / 19
The empty set is called the impossible event. : Let S be a sample space. The event S is called the certainty event. : Two events E and F are said to be mutually exclusive if the sets are disjoint. That is E F =. : Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 14 / 19
Continuous Sample Space Note that the events that we have listed so for we were discrete. But sometimes the event that we want to list can be infinite or would be hard to write with the previous methods. A hospital is carefully measures the length of every baby born. What is a sample space for this experiment? Consider the previous example. (a) Describe the event that the baby is longer than 22 inches. (b) the baby is 20 inches or shorter. (c) the baby is between 19.5 and 21 inches long Solution Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 15 / 19
Section 1.4- Basics of Probability First consideration is going to be sample spaces for which the outcomes are equally likely. We will refer to such sample spaces as uniform sample spaces. flipping a fair coin, throwing a fair die. If S is a finite uniform sample space and E is any event, then the probability of E is given by P(E) = Number of elements in E Number of elements in S = n(e) n(s). Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 16 / 19
A fair die is tossed. (a) What is the sample space S? (b) What is the probability that an even number is observed? (c) What is the probability that 4 or 5 is observed? (d) What is the probability that a number less than 5 is observed? Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 17 / 19
Empirical Probability In some everyday situations, guessing would be difficult. Therefore, we rely on experiments that have been done before and we use these results to find the probabilities of the events. A non-fair die is tossed 1000 times and results were recorded as follows. Outcome 1 2 3 4 5 6 Number Observed 161 179 148 177 210 125 (a) Find the empirical probability that a 3 will occur. (b) Find the empirical probability that a 4 or 5 will occur. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 18 / 19
Probability Distribution Tables Write the probability distribution table for the number of heads when a coin is flipped twice. Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 19 / 19