Basic Probability PDF

Basic Probability Introduction Sample Spaces and Events  Probability Definition 2.1 Probability – is a loosely defined term employed in everyday conversation to indicate measure of one’s belief in the occurrence of a future event. It is a number between 0...

Basic Probability Introduction Sample Spaces and Events  Probability Definition 2.1 Probability – is a loosely defined term employed in everyday conversation to indicate measure of one’s belief in the occurrence of a future event. It is a number between 0 and 1 inclusive associated with the likelihood of occurrence of a given event. 2  Probability Probability is a number that describes the chance of an event occurring if an activity is repeated over and over. A probability of zero means the event cannot occur while a probability of one means the event must occur. Otherwise the probability can be expressed as a fraction, decimal or percent. 3  Probability Experiment An experiment is the process by which an observation or measurement is obtained. Generally, an experiment suggests a planned or carefully controlled laboratory testing situation. 4 Probability Experiment Definition 2.2 However for the purpose of this course, we will be concerned with The term experiment will be used in a broad sense to mean an observation of any physical occurrence. It may refer to any situation of interest whose outcome is determined by chance. 5 Probability Experiments Whenever we manipulate or make an observation of our environment with an uncertain outcome, we have conducted a probability experiment. Examples: Taking an exam Tossing a fair coin. Rolling a fair die. Delivering a sales pitch Buying a cellphone that is defective. Preference for a cola brand. Choosing a number from a set. Examining a fuse for quality control. 6 Sample Space Definition 2.3 The set of all possible outcomes of an experiment is called the sample space for the experiment. The outcomes in the sample space are called sample points. 7 Example – Toss a Coin Twice We could choose to record the sequence of heads (H) and tails (T), then S= {HH, HT, TH, TT} 8 Exercise (Sample Spaces) Determine the sample space of the following experiments: 1. Toss a coin three times Therefore, total numbers of outcome are 23 = 8 9 Exercise (Sample Spaces) Determine the sample space of the following experiments: 2. Rolling two dice Rolling 2 dice gives a total of 36 possible outcomes. 10  Exercise (Sample Spaces) Determine the sample space of the following experiments: 3. Record the sex of successive children in a three-child family. There are 8 possible outcomes, all equally likely to occur. 11  Exercise (Sample Spaces) Determine the sample space of the following experiments: 4. Choosing a card from a standard deck There are 52 ways to draw a card from the standard deck. 12 Exercise (Sample Spaces) Determine the sample space of the following experiments: 5. Taking a 10 question survey with 2 choices (yes or no) for each question: There are 2 ways to answer each question. There are 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 1024 ways to complete the survey. 13 Events Definition 2.4 Events are sets. An event, E, is a subset of the sample space and is denoted by 𝐸 ⊆ 𝑆 An event E is said to occur if the outcome of an experiment is an element of E. Events are classified as either simple or compound. 14 Events Experiment: Toss a die once and record the number on the top face. The sample space, S = {1, 2, 3, 4, 5, 6} Some events associated with this experiment: Getting a one E1={1} Observing a five E2={5} Getting an odd number E3={1,3,5} Observing a number greater than 4 E4={5,6} NOT observing a two E5={1,3,4,5,6} 15 Simple Events vs. Compound Events Definition 2.5 A compound event is any event that can be decomposed into other events. E3 , E4 and E5 are compound events A simple event cannot be decomposed. E1 and E2 are simple events 16 Probability of an Event Definition 2.5 If an experiment can result in any one of k equally likely outcomes and if an event E contains m sample points, then the probability of the event E is 𝑚 # 𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒 𝑝𝑜𝑖𝑛𝑡𝑠 𝑖𝑛 𝐸 𝑛(𝐸) 𝑃 𝐸 = = = 𝑘 # 𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒 𝑝𝑜𝑖𝑛𝑡𝑠 𝑖𝑛 𝑆 𝑛(𝑆) 17 Probability of an Event Example 1. Consider the experiment of tossing a fair die and observing the number on the top face. Find the probability of (a)The event E: an even number results (b)The event G: a number divisible by 3 is tossed 18 Solutions for #’s 1 and 2 1. 𝐿𝑒𝑡 𝑆 = 1, 2, 3, 4, 5, 6. a. E: an even number results 𝐸 = 2, 4, 6 3 1 𝑃 𝐸 = = 6 2 b. E: a number divisible by 3 is tossed 𝐸 = 3, 6 2 1 𝑃 𝐸 = = 6 3 19 Example 3 A balanced coin is tossed three times. 1. Let E1 be the event that you observe at least two heads. What is P(E1)? 2. Let E2 be the event that you observe at exactly two heads. What is P(E2)? 3. Let E3 be the event that you observe at most two heads. What is P(E3)? 20  Solutions for # 3 𝑆 = 𝐻𝐻𝐻, 𝐻𝐻𝑇, 𝐻𝑇𝐻, 𝑇𝐻𝐻, 𝑇𝑇𝐻, 𝑇𝐻𝑇, 𝐻𝑇𝑇, 𝑇𝑇𝑇 1. 𝐸1 = observe at least two heads 𝐸1 = 𝐻𝐻𝐻, 𝐻𝐻𝑇, 𝐻𝑇𝐻, 𝑇𝐻𝐻 4 1 𝑃(𝐸1 ) = = 8 2 2. 𝐸2 = observe exactly two heads 𝑃(𝐸2 ) = 𝐻𝐻𝑇, 𝐻𝑇𝐻, 𝑇𝐻𝐻 3 𝑃(𝐸2 ) = 8 3. 𝐸3 = observe at most two heads 𝑃(𝐸3 ) = 𝐻𝐻𝑇, 𝐻𝑇𝐻, 𝑇𝐻𝐻, 𝑇𝑇𝐻, 𝑇𝐻𝑇, 𝐻𝑇𝑇, 𝑇𝑇𝑇 7 𝑃(𝐸3 ) = 8 21