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Drawing a Tree Diagram You can use a tree diagram to display and count possible choices. Example: A school team sells caps in two colors (blue or white), two sizes (child or adult), and two fabrics (cotton or polyester) cotton Each branch of the “tree” represents one choice – for example, blue-child-cotton child polyester blue cotton adult polyester cotton child polyester There are 8 possible cap choices white cotton adult Copyright © Ed2Net Learning, Inc. polyester

Your Turn! Suppose the caps in the previous example also came in black. Draw a tree diagram. How many cap choices are there? There are 12 possible cap choices Copyright © Ed2Net Learning, Inc.

Counting Principle Another way to count choices is to use the Counting Principle. If there are m ways of making one choice, and n ways of making a second choice, then there are m . n ways of the first choice followed by the second. The Counting principle is particularly useful when a tree diagram would be too large to draw. The Counting Principle is sometimes called the “Multiplication Counting Principle”. Copyright © Ed2Net Learning, Inc.

Using the Counting Principle
Example: How many two-letter monograms are possible? first letter second letter monograms Possible choices possible choices possible choices 26 x 26 = 676 There are 676 possible two-letter monograms Copyright © Ed2Net Learning, Inc.

Your Turn! How many three-letter monograms are possible? 26 x 26 x 26 = 17,576 Copyright © Ed2Net Learning, Inc.

Theoretical Probability
You can count outcomes to help you find the theoretical probability of an event in which outcomes are equally likely. P(event) = number of favorable outcomes number of possible outcomes A sample space is a list of all possible outcomes. Copyright © Ed2Net Learning, Inc.

Using a Tree Diagram Example: Use a tree diagram to find the sample space for tossing two coins. Then find the probability of tossing two tails. heads The tree diagram shows there are four possible outcomes, one of which is tossing two tails. heads tails heads tails P(event) = number of favorable outcomes number of possible outcomes P(two tails) = number of two-tail outcomes = ¼ The probability of tossing two tails is ¼ . tails Copyright © Ed2Net Learning, Inc.

Using the Counting Principle
Example: In lotteries the winning number is made up of four digits chosen at random. Suppose a player buys two tickets with different numbers. What is the probability that the player has a winning ticket? First find the number of possible outcomes. For each digit there are 10 possible outcomes, 0 through 9. Then find the probability when there are two favorable outcomes. P(winning ticket) = number of favorable outcomes = number of possible outcomes 10,000 The probability is 2/10,000 or 1/5,000 1st digit Outcomes 10 total Outcomes 10,000 2nd digit Outcomes 10 3rd digit Outcomes 10 4th digit Outcomes 10 = x x x Copyright © Ed2Net Learning, Inc.

Your Turn! A lottery uses five digits chosen at random. Find the probability of buying a winning ticket. 10 x 10 x 10 x 10 x 10 = 100,000 Copyright © Ed2Net Learning, Inc.

Independent Events Independent Events are events for which the occurrence of one event does not effect the probability of the occurrence of the other. Suppose there are 10 cards with one number from 1 to 10 on them. You are interested to draw an even number and then again a second card with even number. If you replace your first card, the probability of getting an even number on the second card is unaffected. Probability of Independent Events For two independent events A and B, the probability of both events occurring is the product of the probabilities of each event occurring. P(A, then B) = P(A) x P(B) Copyright © Ed2Net Learning, Inc.

Finding Probability of Independent Events
Example: You roll a number cube once. Then you roll it again. What is the probability that you get 2 on the first roll and a number greater than 4 on the second roll? P(2) = 1/6 P(greater than 4) = 2/6 P(2, then greater than 4) = P(2) x P(greater than 4) = 1/6 x 2/6 = 2/36, or 1/18 The probability is 1/18. There is one 2 among 6 numbers on a number cube. There are two numbers greater than 4 on a number cube. Copyright © Ed2Net Learning, Inc.

Your Turn! You toss a coin twice. Find the probability of getting two heads. ½ x ½ = ¼ Copyright © Ed2Net Learning, Inc.

Example Under the best conditions, a wild bluebonnet seed has 20% probability of growing. If you select two seeds at random, what is the probability that both will grow, under best conditions? P(a seed grows) = 20% or 0.20 P(two seeds grow) = P(a seed grows) x P(a seed grows) = 0.20 x 0.20 = 0.04 = 4% The probability that two seeds grow is 4% Copyright © Ed2Net Learning, Inc.

Your Turn! Chemically treated bluebonnet seeds have a 30% probability of growing. You select two such seeds at random. What is the probability that both will grow? 0.3 x 0.3 = 0.09 = 9% Copyright © Ed2Net Learning, Inc.

Dependent Events Dependent Events are events for which the occurrence of one event affects the probability of the occurrence of the other. Suppose you want to draw two even-numbered cards from cards having numbers from 1 to 10. You draw one card. Then , without replacing the first card, you draw a second card. The probability of drawing an even number on the second card is affected. Probability of Dependent Events For two dependent events A and B, the probability of both events occurring is the product of the probability of the first event and the probability that, after the first event, the second event occurs. P(A, then B) = P(A) x P(B after A) Copyright © Ed2Net Learning, Inc.

Finding Probability for Dependent Events
Example: Three girls and two boys volunteer to represent their class at a school assembly. The teacher selects one name and then another from a bag containing the five students’ names. What is the probability that both representatives will be girls? P(girl) = 3/5 P(girl after girl) = 2/4 P(girl, then girl) = P(girl) x P(girl after girl) = 3/5 x 2/4 = 6/20 or 3/10 The probability that both representatives will be girls is 3/10. Three of five students are girls. If a girl’s name is drawn, two of the four remaining students are girls. Copyright © Ed2Net Learning, Inc.

Your Turn! In the previous example, find P(boy, then girl) 2/5 x ¾ = 6/20 = 3/10 P(girl, then boy) 3/5 x 2/4 = 6/20 = 3/10 Copyright © Ed2Net Learning, Inc.

Assessment You can choose a burrito having one filling wrapped in one tortilla. Draw a tree diagram to count the number of burrito choices. Tortillas: flour or corn; fillings: beef, chicken, bean, cheese, or vegetable 10 choices beef beef chicken chicken bean bean flour corn cheese cheese vegetable vegetable Copyright © Ed2Net Learning, Inc.

Assessment There are four roads from Marsh to Taft and seven roads from Taft to Polk. Use the Counting Principle to find the number of routes from Marsh to Polk through Taft. 28 routes Copyright © Ed2Net Learning, Inc.

Assessment Use a tree diagram to find the sample space for tossing three coins. Then find the probability: P(three heads). 1/8 Use the Counting Principle to find the probability of choosing the three winning lottery numbers when the numbers are chosen at random from 1 to 50. Numbers can repeat. 1/125,000 Find the probability. You roll two odd numbers and pick a vowel (when you roll two number cubes and pick a letter of the alphabet at random). 5/104 Copyright © Ed2Net Learning, Inc.

Assessment You roll a number cube twice. What is the probability that you roll 6, then 2 or 5.2/36, or 1/18 Weather forecasters are accurate 91% of the time when predicting precipitation for the day. What is the probability that a forecaster will make correct precipitation predictions two days in a row? About 83% You select the card, G. Then without replacing the card, you select R or A. Find the probability. 4/90 or 2/45 P R E A L G E B R A Copyright © Ed2Net Learning, Inc.