Math 6, Chapter 12: Probability Notes. Probability

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1 Math 6, Chapter 12: Probability Notes Probability Objective: (7.1)he student will perform simple experiments to determine the probability of an event. (7.2)he student will represent experimental results using, fractions, decimals, and percents. Probability is the measure of how likely an event is to occur. hey are written as fractions or decimals from 0 to 1. Probability may be written as a percent, 0% to 100%. he higher the probability, the more likely an event is to happen. For instance, an event with a probability of 0 will never happen. If you have a probability of 100%, the event will always happen. An event with a probability of 1 2 or 50% has the same chance of happening as not happening. Example: ow likely is it that a coin tossed will come up heads? his means that there is as likely a chance of heads as not heads. Example: he weather report gives a 75% chance of rain for tomorrow. his means that there is a likely chance of rain (75%) and an unlikely chance of no rain (25%). Outcomes are the possible results of an experiment. Example: ossing a coin, the possible outcomes (results) are a head or a tail. heoretical probability is based on knowing all the equally likely outcomes of an experiment, and it is defined as a ratio of the number of favorable outcomes to the number of possible outcomes. Mathematically, we write: number of favorable outcomes probability = number of possible outcomes or success probability = success + failure Math 6 Notes Unit 12: Probability Page 1 of 5

2 Example: Suppose you pick a marble from a hat that contains three red, two yellow and one blue marble. What is the probability you draw a yellow marble? # of yellow marbles P = total number of marbles 2 1 P = or 6 3 he probability found in the above example is an example of theoretical probability. Experimental probability is based on repeated trials of an experiment. Example: In the last thirty days, there were 7 cloudy days. What is the experimental probability that tomorrow will be cloudy? 7 P = 30 Odds Note to CCSD teachers: odds is not a concept tested on the CR at the 6 th grade level. hese notes are for your general knowledge. Odds: the ratio of favorable outcomes to the number of unfavorable outcomes, when all outcomes are equally likely. Odds in favor = Odds against = Number of favorable outcomes Number of unfavorable outcomes Number of unfavorable outcomes Number of favorable outcomes Example: Suppose you pick a marble from a hat that contains three red, two yellow and one blue marble. What are the odds (in favor) you draw a yellow marble? 2 of yellow marbles 2 1 P = = = or 1:2 # of marbles not yellow 4 2 Math 6 Notes Unit 12: Probability Page 2 of 5

3 ree Diagrams and he Fundamental Counting Principle Objective: (7.3)he student will determine the outcome for a specific event through the use of sample spaces and tree diagram. ree Diagrams A tree diagram makes it easier to see (count) the number of possible outcomes for experiments when the numbers are small and there are multiple events. o draw a tree diagram, you: 1) begin with a point; then you draw a line for each outcome in the first event. 2) draw lines for subsequent outcomes based on the outcomes from the first event. Example: Draw a tree diagram to show the outcomes for flipping two coins. Start with a point. here are two outcomes for the first coin, a head () or a tail (). Draw lines and label and. For either of the outcomes in the first flip, the second coin could be a head or a tail. So the tree diagram would look like this. Now reading down the tree diagram, the possible outcomes are,,, or. here are 4 possible outcomes when flipping two coins. Extend the tree diagram for three coins. ow many outcomes are there? Math 6 Notes Unit 12: Probability Page 3 of 5

4 Example: Draw a tree diagram to determine the number of different outfits that could be worn if you had two pairs of pants and three shirts. Starting with a point, you have 2 pairs of pants (P1 and P2). For each pair of pants, you have three shirts (S1, S2 and S3) to choose from. P1 P2 S1 S2 S3 S1 S2 S3 here are 6 possible outcomes: P1S1, P1S2, P1S3, P2S1, P2S2, and P2S3. Fundamental Counting Principle ree diagrams are useful to get a picture of what is occurring, but with a large number of events, the tree can get out of hand in a hurry. A quick way to determine the number of possible outcomes in a tree diagram is to multiply the number of outcomes in each event. With the first example using 2 coins, there are 2 outcomes when you flip the first coin and two outcomes when you flip the second coin. 2 2= 4. In the last example, we choose from 2 pairs of pants, then from three shirts. Notice the total number of outcomes we identified using the tree diagram was 6 and 2 3= 6. hose examples lead us to the following generalization: Fundamental Counting Principle: If one event can occur in m ways, and for each of these ways a second event can happen in n ways, then the number of ways that the two events can occur is m n. Example: ow many possible outcomes are there if you roll two cubes with the numbers one through six written on each face? here are 6 outcomes on the first cube, 6 outcomes on the second cube, so using the Fundamental Counting Principal we have = outcomes. Example: ow many possible outcomes are possible for tossing a coin and rolling a cube with the numbers one through six written on each face? Math 6 Notes Unit 12: Probability Page 4 of 5

5 here are two things that can happen when tossing a coin. here are six things that can happen when rolling the cube. Using the Fundamental Counting Principle, we have 2 6= 12outcomes. Example: ow many possible answers are there to a 10 question rue-false test? Using Fundamental Counting Principal, = 2 = Let s extend this to finding the probability of compound events (an event made up of two or more separate events). If the occurrence of one event does not affect the probability of the other, the events are independent. Probability of Independent Events = P(A) i P(B) Example: An experiment consists of flipping a coin 2 times. What is the probability of flipping heads both times? he flip of a coin does not affect the results of the other flips, so the flips are independent. For each flip, P() = So P(, ) = i or Example: You have 3 colors of t-shirts (red, blue, green) 2 colors of shorts (white, black) from which to choose. What is the probability of randomly choosing a blue shirt with black pants? For the choice of shirt, P() = 1 3, for the shorts P(S) = ; So P(, S) = i or Math 6 Notes Unit 12: Probability Page 5 of 5