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Chapter 7: Probability Lesson 1: Basic Principles of Probability Mrs. Parziale Vocabulary • Experiment: a situation that has several possible results • Outcomes: results of an experiment • Sample space: the set of all possible outcomes • Event: subset of the sample space • Relative Frequency: comes from conducting an experiment – is your results from a simulation. In contrast, probability uses information to predict future events. More Terms • Probability of an event: a number from 0 to 1 that measures its likelihood. Notation: N(E) = the number of elements in a set E S = sample space P(E) = the probability that E occurs Assuming each outcome in S is equally likely N ( E ) # of Outcomes In Event Formula: P(E) = # of Outcomes In Sample Space S Theorem: Basic Properties of Probability: • If S = sample space and P(E) be the probability of event E, then • (i) 0 P ( E ) 1 • (ii) If E = S, then P(E) = 1 • (iii) If E = {} (empty set), then P(E) = 0. Example 1: Toss a fair coin 2 times. Find the probability of getting 2 heads: • Sample space: S = { HH, TH, HT, TT } (S = 4) • Event: 2 heads • N(E) = # of outcomes = 1 (HH) P(2 heads) = N ( E ) 1 S 4 Example 2: Toss a fair coin 3 times. Find the probability of getting exactly one head: Sample space: S= Event: • N(E) (# of outcomes) = • P(E) = Example 3: Roll two die. Take the sum of the numbers on top. (a) Event: sum of 10 P(E) = (b) Event: sum less than 5 P(E) = (c) Event: sum is less than 50 P(E) = (d) Event: sum is equal to 1 P(E) = (e) What sum has the highest probability of occurring? Example 4: A researcher is studying the number of boys and girls in families with three children. Assume that the birth of a boy or a girl is equally likely. a. Find the probability that a family of three children has exactly one boy. First identify the sample space. b. Find the probability that a family of three children has exactly two girls. Example 5: Students were testing the theoretical probability of getting heads or tails when tossing a coin. While performing an experiment of 2500 tosses, heads came up 2466 times. a. What is the probability that the 2501st coin tossed will be heads? b. What is the probability that the 2501st coin tossed will be tails? Closure • Probability of an event can be expressed as a number between _______ and _______ , inclusive. • If the event equals the sample space, P(E) = __ • A probability of zero means ______________ • What is the difference between Probability and Relative Frequency?