Probability Theoretical Probability Finding the complements of an Event Experimental Probability Tree Diagrams Counting Principle Compound Events

Theoretical Probability Remember: And probability can be written as fractions, percents, and decimals Example: Suppose you select a letter at random from the letters shown at the right. What is the probability that you select a vowel?

Complements of an Event The complement of an event is the collection of outcomes not contained in the event. In other words, the complement is the opposite of the probability of an event occurring. It is written : P(not event) Say we have a 4 color spinner and want to know the theoretical probability of spinning a blue…it would ¼ , 25%, or .25 The complement of not spinning a blue would be ¾ , 75%, or .75 The rule is: For any event A, the complement is not A And P(A) + P(not A) = 1 Or P(not A) = 1 – P(A)

Experimental Probability Remember, experimental probability is found when conducting an experiment and: Example: Suppose you attempt 16 free throws in a basketball game. Your results are at the right. Find the experimental probability of making a free throw.

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Tree Diagrams Remember: A tree diagram is a branching picture that shows all the combinations of choices Example: Suppose you are going to travel on a river. You have two choices of boats—a kayak or a rowboat. The river splits into three smaller streams going north, northwest, and northeast? What are the possible outcomes for your journey? Draw a tree diagram

Exercises: A car can be purchased with either two doors or four doors. You may also choose leather, vinyl, or fabric seats and two choices of color tan or white. Draw a tree diagram that shows all the buying options. B. Two spinners are spun. Each spinner has one half red and the other half blue. Create a tree diagram to find all of the possible outcomes. C. A penny is tossed, and a number cube is rolled. Use a tree diagram to find all the possible outcomes.

Counting Principle: Do you see a pattern? In the example of tree diagrams, there are 2 choices of boats and 3 choices of direction. How many choices are there in all? 6 For the car there are 2 choices of doors, and 3 choices of seats, and 2 choices of color. How many choices in all? 12 For the 2 spinners, there are 2 equal sections. How many outcomes are possible? 4 For the penny and die, there are 2 sides on the penny, and 6 sides on the die. How many outcomes are possible? Do you see a pattern?

The Counting Principle If there are m ways to make one choice and n ways to make a second choice, then there are m x n outcomes. Example: If you choose a shirt in 5 sizes and 7 colors, then you can choose m x n, or 5 x 7 = 35 shirts Suppose you order a sandwich by choosing one bread and one meat from the menu. How many different sandwich are available?

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Compound Events What you will learn: To find the probability of independent events To find the probability of dependent events What is the probability of rolling a 6 and then spinning a blue?

Exploring Multiple Events Suppose you select a bead from the beads above. You then select another bead. The selection of two beads involves two events. A compound event consists of two or more events.

Compound Events A compound event consists of events that either do or do not depend on each other. Two events are independent events if the occurrence of one event does not affect the probability of the occurrence of the other. Example: Suppose a family has two children. The gender of the second child is independent of the gender of the first child Probability of Independent Events If A and B are independent events, then P(A, then B) = P(A) x P(B)

Probability of Independent Events A family wants to have two children. What is the probability that both children will be girls? Assume the probability of having a girl is ½

Probability of Independent Events 2. “Spin Your Initials” uses a wheel lettered with equal sections A-Z. Suppose you spin it twice. Find P(B, then Z). The two events are independent. There are 26 letters in the alphabet.

Probability of Independent Events A bag contains 3 blue marbles, 4 red marbles, and 2 white marbles. Three times you draw a marble and return it. What is the probability of P(red, then white, then blue) 4/9 x 2/9 x 3/9 = 24/729 = 3/243 The key to an independent event during drawing, is that the item is replaced before the next item is drawn or selected!

Exercises: at right

Dependent Events Suppose you play a game with cards numbered 1-5. You draw two cards at random. You draw the first card and do not replace it. The probability in the second draw depends on the result of the first draw. Two events are dependent events if the occurrence of one event affects the probability of the occurrence of the other event

Dependent Events To find the probability of an event after one selection: Example: You select a card at random from those below. The card has the letter M. Without replacing the M card, you select a second card. Find the probability of selecting a card with the letter A after selecting M.

Dependent Events Probability of Dependent Events If event B depends on event A, then P(A, then B) = P(A) x P(B after A) To play “Draw your initials,” you draw a card from a bucket that contains cards lettered A-Z. Without replacing the first card, you draw a second one. Find the probability of winning if your initials are C and M. The two events are dependent. After the first selection, there are 25 letters.

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