PROBABILITY Algebra 2

Probability Definition: Theoretical Probability – what should happen mathematically Experimental Probability – what actually happens when you carry out the event The more trials you carry out, the closer experimental probability will approach theoretical probability.

Probability Experiment One Experiment Two Flip the coin 10 times, each time record the outcome (heads or tails). What should you have seen with ten flips? Experiment Two Flip the coin 20 times, each time record the outcome (heads or tails). What should you have seen with twenty flips?

Compound Events If A and B are events, then the probability of A or B is: If A and B are mutually exclusive, then the probability of A or B is:

Compound Events Experiment One: From a deck of cards you will choose “one” card (there are 52 cards in a deck). What is the probability that you get a face card or a heart? How many face cards are there? How many hearts are there? What do you notice? Remember: “One Event = ADDITION”

Compound Events Experiment Two: From a deck of cards you will choose “one” card (there are 52 cards in a deck). What is the probability that you get an Ace or a seven? How many aces are there? How many sevens are there? What do you notice? Remember: “One Event = ADDITION”

The Complement The event called the complement of event A, consists of all outcomes that are not in A. The notation is read as “A prime”. The probability of the compliment of A is

Probability of Independent Events The outcome of A does not effect the outcome of B. “The item is replaced”

Probability of Independent Events Experiment One: What is the probability that you choose two consecutive green coins? You will choose one coin from the cup. What is the probability that it is green? Return it to the cup. You will choose a second coin from the cup. What is the probability that it is green? Return it to the cup. Remember: “Two Events = MULTIPLICATION”

Probability of Dependent Events The outcome of A effects the outcome of B. “The item is NOT replaced”

Probability of Dependent Events Experiment Two: What is the probability that you choose two consecutive green coins? You will choose one coin from the cup. What is the probability that it is green? Keep it out if the cup. You will choose a second coin from the cup. What is the probability that it is green? Remember you have one green coin out. Remember: “Two Events = MULTIPLICATION”

Binomial Probability For a binomial experiment consisting of n trials, the probability of exactly r successes is:

Standard Deviation The steps to calculate the standard deviation are: 1. Calculate the mean of the series. 2. Calculate the differences between the elements and the mean: for all elements. 3. Calculate the squares of the differences. 4. Calculate the mean of the squares by adding all the squares and dividing by the number of elements. 5. Calculate the square root of the mean. This is the standard deviation.

Standard Deviation Example: Calculate the standard deviation of the series {1,3,12,5,9}. Solution: The standard deviation can be calculated as follows: 1. Mean of the series = (1+3+12+5+9)/5 = 30/5 = 6 2. The differences between the individual elements and mean would be: {1-6, 3-6, 12-6, 5-6, 9-6} = {-5,-3,6,-1,3} 3. The squares of the differences would be {25,9,36,1,9}. 4. The mean of the squares would be (25+9+36+1+9)/5 = 80/5 = 16 5. The square root of the mean of squares would be = 4 Thus the standard deviation would be 4.