3-3 Multiple Events 21 October 2010 While I’m gone • Groups of three – Two players, one counter • Play rock-paper-scissors • Best of three wins throws wins game • Count keeps track of who threw what, (including names) and who won each game • Keep playing until I return, switch off roles, if you’d like Independent Events • Several events happening, simultaneously or consecutively. • Independent events – The outcome of an event does not depend on the previous event, nor does it have an effect on the next event – For example, rolling dice, flipping a coin, or picking a card from a deck and replacing same before drawing the next • Dependent events – An event changes the sample space for the next event. – For example, drawing a card and keeping it before the next card. – The probabilities might have to be recalculated for the next event Finding Probability of multiple events • If events A and B are independent, the probability of both events occurring is the product of the individual probabilities. In other words: – P(A and B) = P(A) · P(B) • The probability of flipping heads twice in a row: – P(Heads) = – P(Heads and Heads) = • The probability of rolling two consecutive ‘sevens’ – P(7) = – P(7 and 7) =
3-3 Multiple Events 21 October 2010 Independent events • The probability of having three daughters – P(1 daughter) = – P(Three daughters) = • Acing a multiple choice quiz of five questions each with five choices by guessing? – P(1 correct) = – P(Five correct) = More probability Free throw Field Goal 3 pointer Missed 8 32 10 Made 20 36 4 • Making consecutive 3-point attempts • 10 consecutive free throws • Three different baskets in a row Dependent events • We have jar of 16 coins, 4 • However, once we pick the each of quarters, nickels, first penny, the contents of dimes, and pennies. What is the jar has changed, the probability of picking therefore: two consecutive pennies if • P(Second penny) = we don’t replace the first? • P(First penny) = • Finally, the probability of picking two consecutive pennies is • P(Two pennies) =
3-3 Multiple Events 21 October 2010 Probability of Multiple (Consecutive) Events Live example 1. Three consecutive red 1. Two shapes that are circles with either red or circles replacement w/o replacement 2. Three consecutive red 2. A blue shape followed circles without by a red shape with replacement replacement 3. A blue triangle, red 3. Two triangles w/o circle, blue circle, red replacement triangle without replacement Dependent events Peppermint Spearmint Wintergreen Gum 10 15 11 Lifesaver 12 8 4 • Picking two sticks of gum w/o replacement • Picking three peppermints w/o replacement • Picking two lifesavers or spearmint candies
3-3 Multiple Events 21 October 2010 More live examples 1. What is the probability of drawing two aces from a deck of cards… – If the first ace is replaced – If the first ace isn’t replaced? 2. What is the probability of drawing a hand of five hearts? 3. What is the probability of drawing a five-hand flush (same suit)? 4. P(Two Jacks with replacement) = 5. P(Three of a kind w/o) = 6. P(five red cards w/o replacement) = Probability • The probability of purchasing a defective widget is 2%. If I buy 3 widgets what is the probability that 1. All three are defective? 2. All three are OK 3. At least one is defective More probability • Punxsutawney Phil has predicted 13 early springs and 99 late winters. • P(Early Spring) = • P(Two consecutive early springs) = • Find the probability of at least one early spring in the next four years
3-3 Multiple Events 21 October 2010 • I need to pick a committee of three students from my engineering class. There are seven frosh, five sophomores, three juniors, and 10 seniors. Find the following probabilities 1. All three juniors make up the committee 2. The committee includes a junior, senior, and sophomore
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