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Lesson 33 Finding the probability of independent and dependent events probability • The probability of an event is the number of favorable outcomes divided by the number of total possible outcomes • Number of favorable outcomes • Number of possible outcomes Independent events • Events where the outcome of one does not affect the probability of the other are called independent events. • To find the probability of 2 independent events, multiply the probabilities of the 2 events. • Flipping a coin is an independent event • Spinning a spinner is an independent event • Rolling a dice is an independent event • The result of one does not affect the result of the other examples • You have a coin and a spinner with 5 spaces: • What is the probability of spinning a 3 and landing on heads? • These are independent so: • P(3 and heads) = P(3) x P(heads) • =1 x 1 = 1 • 5 2 10 • What is the probability of spinning a 5 and then a 1? • These are independent events so: • P(5 and 1) = P(5) x P(1) • = 1 x 1 = 1 • 5 5 25 Dependent events • With dependent events, the outcome of one event does affect the probability of the other event. • Sometimes dependent events are described as events without replacement. • Example: drawing a red marble out of a bag and not replacing it. • The 2nd draw is affected by not putting it back in the bag after you draw it out, because the number of possible outcomes changes, and sometimes the number of favorable outcomes also changes. Independent or dependent? • 1) rolling a 6 on a dice and rolling a 4 on another dice • 2) drawing a marble from a bag, putting it back in the bag, and then drawing a blue marble • 3) rolling a 6 on a dice and then rolling a 4 on the same dice • 4) drawing a red marble from a bag, not putting it back, and then drawing a blue marble • 5) tossing 2 coins, a dime and a quarter • 6)putting 10 students' names in a bag and drawing 2 names without replacing the first name drawn Using a tree diagram • 1st 2nd • • H • H HH T HT • • T • H TH T TT outcomes Calculating the probability of dependent events • Joe has 2 squares and 3 circles in a bag. • Find the probability of drawing a circle, keeping it, and then drawing another circle. • P(1st circle) = 3/5 • For the 2nd draw, a circle has been removed, so now there are only 2 circles and only 4 shapes left • So P(2nd circle)= 2/4 • So P(1st circle) x P(2nd circle) = 3 x 1 = 3 • 5 2 10 Find probability • A bag has 5 $10 bills and 15 $1 bills: • What is the probability of drawing a $1 bill, keeping it, and then drawing a $10 bill? • What is the probability of drawing a $10 bill, keeping it, and then drawing another $10 bill? odds • Odds are another way of describing the likelihood of an event. • Odds are expressed as ratios, usually written with a colon. • Odds can be calculated for something or against something happening. Definition of odds • Odds of an event: a ratio expressing the likelihood of an event. • Assume all outcomes are equally likely, and there are m favorable and n unfavorable outcomes. • The sum of the favorable outcomes and the unfavorable outcomes should be the same as the total possible outcomes. • The odds for the event are: m:n • The odds against the event are: n:m Calculating odds • A bag contains 6 red marbles, 2 yellow marbles, and 1 blue marble. • 1) What are the odds of drawing a red marble? • There are 6 favorable outcomes (6 red marbles • There are 3 unfavorable outcomes (3 not red) • So odds are: 6:3 or 2:1 odds • 1) What are the odds of drawing a blue marble if there are 6 red, 2 yellow and 1 blue marbles. • 2) What are the odds against drawing a blue marble? problem • The computer club has 10 boys and 12 girls. The chess club has 15 boys and 5 girls. Each club will choose a member at random to be club treasurer. • 1) What is the probability that both treasurers will be girls? • 2) Both girls resign and a different pair of students is chosen. What is the probability that both treasurers will again be girls?