# PROBABILITY. SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a number between 0 and 1.

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1 PROBABILITY SIMPLE PROBABILITY SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a number between 0 and. There are two categories of simple probabilities. THEORETICAL PROBABILITY is calculated probability. If every event is equally likely, it is the ratio of the number of ways the event can occur to the total number of possible outcomes. theoretical probability = number of ways to get want you want total number of possible outcomes EXPERIMENTAL PROBABILITY is the probability based on data collected in experiments. experimental probability = number of times the event occurred total number of outcomes Example There are three pink pencils, two blue pencils, and one green pencil. If one pencil is picked randomly, what is the theoretical probability it will be blue? Find the total number of possible outcomes, that is, the total number of pencils. + + = 6 Find the number of specified outcomes, that is, how many pencils are blue? Find the theoretical probability. P(blue pencil) = 6 =. (You may reduce your answer.)

2 Example Jayson rolled a die twelve times. He noticed that three of his rolls were fours. a. What is the theoretical probability of rolling a four? Because the six sides are equally likely and there is only one four, P() = 6. b. What is the experimental probability of rolling a four? There were three fours in twelve rolls. The experimental probability is: P() = =. Example On the spinner, what is the probability of spinning an A or a B? The probability of an A is. The probability of a B is. Add the two probabilities for the combined total. + = = P(A or B) = A C B D Example What is the probability of spinning red or white? We know that P(R) = and P(W) =. Add the probabilities together. P(R or W) = 7 + = 7 Red White Blue

3 Problems. There are five balls in a bag: red, blue, and white. What is the probability of randomly choosing a red ball?. In a standard deck of cards, what is the probability of drawing an ace?. A fair die numbered,,,,, 6 is rolled. What is the probability of rolling an odd number?. In the word "probability," what is the probability of selecting a vowel?. Anna has some coins in her purse: quarters, dimes, nickels, and pennies. a. What is the probability of selecting a quarter? b. What is the probability she will select a dime or a penny? 6. Tim has some gum drops in a bag: 0 red, green, and yellow. a. What is the probability of selecting a green? b. What is the probability of not selecting a red? 7. What is the probability of spinning: a. pink or blue? b. orange or pink? c. red or orange? d. red or blue? 0 red orange pink blue. What is the probability of spinning: a. A or C? b. B or C? c. A or D? d. B or D? e. A, B, or C? A B 6 C D Answers.... a. =. There are odd numbers. 6 = b. 7 = 6. a. 7 b a. 0 b. 0 c. 7 0 d a. b. 7 c. 7 d. e.

4 DEPENDENT AND INDEPENDENT EVENTS Two events are DEPENDENT if the outcome of the first event affects the outcome of the second event. For example, if you draw a card from a deck and do not replace it for the next draw, the two events drawing one card without replacing it, then drawing a second card are dependent. Two events are INDEPENDENT if the outcome of the first event does not affect the outcome of the second event. For example, if you draw a card from a deck but replace it before you draw again, the two events are independent. Example Aiden pulls an ace from a deck of regular playing cards. He does not replace the card. What is the probability of pulling out a second ace? First draw: Second draw: aces left cards left to pull from This is an example of a dependent event the probability of the second draw has changed. Example Tharon was tossing coins. He tossed a head. What is the probability of tossing a second head? The probability for the second event has not changed. This is an independent event.

5 Problems. You throw a die twice. What is the probability of throwing a six and then a second six? Are these independent or dependent events?. You have a bag of candy filled with pieces of the same size and shape. Four are gumballs and six are sweet and sours. You draw a gumball out, decide you don't like it, put it back, then select another piece of candy. What is the probability of selecting another gumball? Are these independent or dependent events?. Joey has a box of blocks with eight alphabet blocks and four plain red blocks. He gave an alphabet block to his sister. What is the probability his next selection will be another alphabet block? Are these independent or dependent events?. In your pocket you have three nickels and two dimes. a. What is the probability of selecting a nickel? b. What is the probability of selecting a dime? c. If you select a nickel and place it on a table, what is the probability the next coin selected is a dime? Are these independent or dependent events? d. If all the coins are back in your pocket, what is the probability that the next coin you take out is a dime? Is this event independent or dependent in relation to the previous events?. How do you tell the difference between dependent and independent events? Answers., independent (the probability doesn't change) 6. 0 or, independent. 7, dependent. a. b. c. =, dependent d., independent. For dependent events, the second probability changes because there is no replacement.

6 COMPOUND PROBABILITY COMPOUND PROBABILITY When multiple outcomes are specified (as in problem 7(a) in the first section, pink or blue), and either outcome may occur but not both, find the probability of each specified outcome and add their probabilities. If the desired outcome is a compound event, that is, it has more than one characteristic (as in Example, an I and a T), find the probability of each outcome (in Example, the probability of "I," then the probability of "T") and multiply the probabilities. Example To find the P(I and T): Spinner # Spinner # Find the probability of I:. Find the probability of T:. I U A T F Multiply them together.! = P(I and T) = P(I) = P(U) = 6 P(A) = P(T ) = P(F) = This can also be shown with an area diagram. Divide the top of a square proportionally to correspond with the areas of the sectors in spinner #. Divide the side of the square proportionally to correspond with the areas of the sectors in spinner #. Drawing the created interior rectangles yields all possible outcomes for the two spinners. Multiplying the dimensions of the interior rectangles determines its area and also the probability of the compound event. Spinner # T Spinner # I A ( ) U ( ) ( ) ( ) IT AT UT F ( ) IF AF UF 6 The P(I and T) = area of rectangle IT =! = Area diagrams will be especially useful in solving conditional probability problems, which are covered in the final section of this probability practice.

7 Example What is the probability of spinning a black dog or a black cat? P(BCat) = = P(BDog) = = Black White Cat Dog Goat P(BCat or BDog) = + = + = Problems. If each section in each spinner is the same size, what is the probability of getting a black truck? Red Blue Black Green Car Truck. Bipasha loves purple, pink, turquoise and black, and has a blouse in each color. She has two pair of black pants and a pair of khaki pants. If she randomly chooses one blouse and one pair of pants, what is the probability she will wear a purple blouse with black pants?. The spinner at right is divided into five equal regions. The die is numbered from to 6. What is the probability of: Red White a. getting red and rolling a? b. white or blue and an even number? Yellow Green Blue

8 . What is the probability Joanne will win a: a. chocolate double scoop? Choc. Vanilla Single Scoop Soda b. chocolate or strawberry sundae? c. chocolate double scoop or chocolate sundae? Strawberry Double Scoop Sundae. Jay is looking in his closet, trying to decide what to wear. He has red t-shirts, black t-shirts, and white t-shirts. He has pair of blue jeans and pair of black pants. a. What is the probability that he randomly chooses a red shirt with jeans? b. What is the probability that he randomly chooses an all black outfit? c. Which combination out of all his possible choices has the greatest probability of being randomly picked? How can you tell? 6. What is the probability of spinning a: a. Red or Green? b. Blue or Yellow? c. Yellow or Green? G B 6 R Y Answers. =. = = 6. a. 6 = 0 b. ( + ) 6 =. a. =. a. 7 = 6 6. a. c. 7 b. ( + ) = 6 c. + = 6 b. 7 = c. white shirt, blue jeans; most selections in both categories b.

9 CONDITIONAL PROBABILITY Conditional probability is the probability that a second event occurred knowing that a first event occurred. The probability that event A occurred knowing that event B occurred is written P(A B) and is verbalized "the probability of A, given B." Example Two dice are thrown. What is the probability of getting 0 points? Since all possibilities are equally likely we could just make a list of ways to make 0 points: (, 6), (, ) and (6, ). Three ways out of 6 possible outcomes is. If we knew we made 0 points what was the probability that each die showed five points? Of those three ways, only one has two-fives so the conditional probability is. It can also be viewed as an area diagram where each cell has area 6. P(0) = = 6 = P(two-fives 0) = 6!out!of 6 =. 6 X X 6 X Example A spinner comes up red % of the time and green % of the time. The rest of the time it lands on blue. Draw an area diagram for spinning twice, and shade the region on your area diagram corresponding to getting the same color twice. See diagram at right. Since the probabilities of each color are not the same, just making a list will not work. What is the probability that both spins land on the same color? Add the areas or probabilities of the shaded squares: + + =. 6 6 P(B)=/ P(B) =/ / P(R) =/ P(G) =/ If you know that you got the same color twice, what is the conditional probability the color was blue? The possible region is of the area. The desired area is of the region. out of means = =. 6 P(R)=/ P(G)=/ /6 /6

10 Problems. Two dice are thrown and the sum is seven points. a. Shade the squares on an area diagram where this outcome could occur. b. What is the probability of getting seven points? c. If the total was seven points, what is the probability that one of the dice was a six?. Suppose you roll two dice and the sum is more than. a. Shade the squares on an area diagram where this outcome could occur. b. What is the probability that both dice show the same number? c. What is the probability that exactly one 6 is showing? d. What is the probability that at least one is showing?. A spinner comes up blue, red, and green with a probability of for each color. a. Sketch an area diagram for spinning twice. b. Shade the region on your area diagram that corresponds to landing on the same color twice. c. What is the probability that both spins land on the same color? d. If you know that you got the same color twice, what is the probability the color was red?. Another spinner. This time the spinner lands on red half of the time and on green one-third of the time. The rest of the time it lands on blue. a. Draw an area diagram for spinning twice and shade the region that corresponds to getting the same color on both spins. b. Suppose you know that the spinner landed on the same color twice. What is the probability that color was green?

11 . In the children's game Build-a-Farm each player first spins a spinner. Half of the time the spinner comes up red. Half of the time the spinner comes up blue. If the spinner is red, you reach into the red box. If the spinner is blue, you reach into the blue box. The red box has 0 chicken counters, 0 pig counters, and 0 cow counters, while the blue box has chicken counters, pig counters and cow counter. a. Sketch an area diagram for the situation where a child spins and then draws. Note that the parts corresponding to the two boxes of animal markers will be quite different. b. Shade the parts of the diagram corresponding to getting a pig counter. What is P(pig)? c. Find the probability of getting a cow counter. d. Find the probability that if you got a cow counter you also spun red. e. Find the probability that if you got a cow counter you also spun blue. 6. A spinner has two colors, red and blue. The probability the spinner will land on blue is x. a. What is the probability it will land on red? b. Sketch an area diagram for spinning twice. c. If the spinner is spun twice, what is the probability it will land on the same color both times? d. Given that the spinner lands on the same color twice, what is the probability that it landed on blue both times?

12 Answers 6 6 a. b. = c = a. 6 b. = c. 6 = d P(B) P(R) P(G). a/b. c. d. P(B) 0 = P(R) P(G) P(R) P(G) P(B) a. b. 7 P(R)! = P(G)! = 9 P(B) 6! 6 = 6

13 a. b c. + = d e ( ) = 6. a.! x b. P(red) - x P(blue) x P(red) - x (! x) x(! x) = x! x P(blue) x x(! x) = x! x x c. d. x + (! x) = x! x + x x! x +

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