Applicable Mathematics “Probability” Page 113

Definitions Probability is the mathematics of chance. It tells us the relative frequency with which we can expect an event to occur The greater the probability the more likely the event will occur.

Definitions Certain Impossible /50 Probability is the numerical measure of the likelihood that the event will occur. Value is between 0 and 1, inclusively. Sum of the probabilities of all events is 1.

Definitions A probability experiment is an action through which specific results (counts, measurements, or responses) are obtained. The result of a single trial in a probability experiment is an outcome. The set of all possible outcomes of a probability experiment is the sample space, denoted as S. e.g. All 6 faces of a die: S = { 1, 2, 3, 4, 5, 6 }

Definitions Other Examples of Sample Spaces may include: Lists Tables Grids Venn Diagrams Tree Diagrams You may use a combination of these Where appropriate always display your sample space

Definitions An event consists of one or more outcomes and is a subset of the sample space. Events are often represented by uppercase letters, such as A, B, or C. Notation: The probability that event E will occur is written P(E) and is read “the probability of event E.”

Definitions The Probability of an Event, E: Consider a pair of Dice Each of the Outcomes in the Sample Space are random and equally likely to occur. e.g. P( ) = (There are 2 ways to get one 6 and the other 4) P(E) = Number of Event Outcomes Total Number of Possible Outcomes in S

Definitions There are three types of probability 1. Theoretical Probability Theoretical probability is used when each outcome in a sample space is equally likely to occur. P(E) = Number of Event Outcomes Total Number of Possible Outcomes in S The Ultimate probability formula

Definitions There are three types of probability 2. Experimental Probability Experimental probability is based upon observations obtained from probability experiments. The experimental probability of an event E is the relative frequency of event E P(E) = Number of Event Occurrences Total Number of Observations

Definitions There are three types of probability 3. Subjective Probability (you do not need to know) Subjective probability is a probability measure resulting from intuition, educated guesses, and estimates. Therefore, there is no formula to calculate it. Usually found by consulting an expert.

Definitions Law of Large Numbers. As an experiment is repeated over and over, the experimental probability of an event approaches the theoretical probability of the event. The greater the number of trials the more likely the experimental probability of an event will equal its theoretical probability.

Counting Principle If there are m ways to do one thing, and n ways to do another, then there are m * n ways of doing both. Let's say that you want to flip a coin and roll a die. There are 2 ways that you can flip a coin and 6 ways that you can roll a die. There are then 2x6=12 ways that you can flip a coin and roll a die.

Probability & Tree Diagrams

For example – a fair coin is flipped twice H H H T T T HH HT TH TT 2 nd 1 st Possible Outcomes 4 possible outcomes Heads then Heads, Heads then Tails, Tails then Heads, Tails then Tails.

Attach probabilities H H H T T T HH HT TH TT 2 nd 1 st ½ ½ ½ ½ ½ ½ P(H,H)=½x½=¼ P(H,T)=½x½=¼ P(T,H)=½x½=¼ P(T,T)=½x½=¼ INDEPENDENT EVENTS – 1 st spin has no effect on the 2 nd spin

Calculate probabilities H H H T T T HH HT TH TT 2 nd 1 st ½ ½ ½ ½ ½ ½ P(H,H)=½x½=¼ P(H,T)=½x½=¼ P(T,H)=½x½=¼ P(T,T)=½x½=¼ Probability of at least one Head? * * *

For example – 10 colored beads in a bag – 3 Red, 2 Blue, 5 Green. One taken, color noted, returned to bag, then a second taken. B RR 2 nd 1 st B B B R R R R G G G G RBRB RGRG BRBR BB BGBG GRGR GBGB GG INDEPENDENT EVENTS

B RR 2 nd 1 st B B B R R R R G G G G RBRB RGRG BRBR BB BGBG GRGR GBGB GG Probabilities P(RR) = 0.3x0.3 = 0.09 P(RB) = 0.3x0.2 = 0.06 P(RG) = 0.3x0.5 = 0.15 P(BR) = 0.2x0.3 = 0.06 P(BB) = 0.2x0.2 = 0.04 P(BG) = 0.2x0.5 = 0.10 P(GR) = 0.5x0.3 = 0.15 P(GB) = 0.5x0.2 = 0.10 P(GG) = 0.5x0.5 = 0.25 All ADD UP to 1.0

Choose a meal Making a Tree Diagram Main course Salad 0.2 Egg & Chips 0.5 Pizza 0.3 Pudding Ice Cream 0.45 Apple Pie 0.55 S E P IC AP IC AP IC AP P(S,IC) = 0.2 x 0.45 = 0.09 P(S,AP) = 0.2 x 0.55 = P(E,IC) = 0.5 x 0.45 = P(E,AP) = 0.5 x 0.55 = P(P,IC) = 0.3 x 0.45 = P(P,AP) = 0.3 x 0.55 = Adds up to 1

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