Probability The Study of Chance!

When we think about probability, most of us turn our thoughts to games of chance When we think about probability, most of us turn our thoughts to games of chance

Probability is also the underlying foundation for the methods we use in inferential statistics. Let’s learn some basics

PROBABILITY BASICS EXPERIMENT: SAMPLE SPACE: EVENT: Any action with unpredictable outcomes The set of all possible outcomes “S” A specific outcome P (A):The probability of outcome A occurring Complement : The complement of any event “A”, consists of all outcomes in which “A” does NOT occur.

PROBABILITY BASICS A probability is a number between 0 and 1 For any event A, 0 ≤ P(A) ≤ 1 The probability of the set of all possible outcomes (sample space) must be 1. P(S)=∑P(A) = 1 This is known as “The something’s gotta happen rule” The probability of an impossible event is 0 The probability of an event that is certain is 1 Most events have probabilities somewhere between!

Approaches to Probability Relative Frequency Approximation: (Experimental Probability)  Conduct the experiment a large number of times and record the results Then:

For Example ??What’s the probability that I roll a 4 with a standard die (A) ?? To answer this using relative frequency approach (experimental probability) I rolled a die 200 times. Outcomes Outcome Frequency

2 nd Approach to Probability Classical Approach to Probability (Theoretical Approach) **Requires Equally-likely Outcomes**  Assume that a procedure has “n” different simple events and each of those events has an equal chance of happening Then:

For Example ??What’s the probability that I roll a 4 with a standard die (A) ?? Let’s answer this using the relative frequency approach (theoretical probability) A standard die has one “4” A standard die has 6 possibilities

So, let’s summarize Experimental P(4) =.145 Theoretical P(4) =.166 Why did we get two different answers? Probability is really about what would happen “in the long run”. The “Law of Large Numbers” tells us that experimental probability will approach the actual (theoretical) probability in a LARGE number of trials. If we increased the number of rolls our probability would get closer and closer to.166

3 rd Approach to Probability Subjective Probability  Probabilities are found by simply guessing or estimating its value based on prior knowledge of the relevant circumstances Examples of subjective probability include forecasting the weather

Complementary Events The complement of an event is when the event does NOT happen. The complement of an event is when the event does NOT happen. Remember that one of our basics of probability is the “something’s gotta happen rule” which says that the sum of all possibilities equals 1. So, the probability of a complement (A’) can be found with 1 – the probability that the event happened. P(A’) = 1 – P(A)

For Example For a single card drawn from a standard deck of cards, what is the probability that it is NOT a diamond? Facts about cards: 13 diamonds 13 hearts 13 spades 13 clubs We can use what we know about complementary events to answer this question. Call “diamond” event A P(A’) = 1 – P(A) P(A’) = 1- (13/52)=.75

Multiple Events Consider the event: Consider the event: Tossing a coin followed by rolling a standard dieTossing a coin followed by rolling a standard die We want to calculate the probability of rolling a 5.We want to calculate the probability of rolling a 5. To find this probability, consider the following model.To find this probability, consider the following model.

Toss a coin We will use a tree diagram to represent the stages in the event. In order to keep the “tree” manageable, we will define each stage as simply as we can. i.e.—”success” or “not a success” The first stage has two possible outcomes Heads Tails The second stage was to roll a die. Although there are 6 possible outcomes, we are only interested in the outcome “roll a 5”. We can simplify our tree by designating only “success”—5, and not a success—5 c 5 5c5c 5 5c5c Remember: Our question is P(rolling a 5) = ? Notice this happens in two places in our tree. To find the probabilities for these two “branches”, we multiply the individual probabilities across and then add the two branches together P(heads & 5) = P(tails & 5) =

Calculating Probability This means that the probability of rolling a “5” is calculated by: This means that the probability of rolling a “5” is calculated by: = (1/2)(1/6) + (1/2)(1/6)= (1/2)(1/6) + (1/2)(1/6) = (1/12) + (1/12)= (1/12) + (1/12) = (2/12)= (2/12)

Multiple Events and a couple of shortcuts Let’s consider the following situation Let’s consider the following situation Roll a standard die 3 timesRoll a standard die 3 times Now find the following probabilities Now find the following probabilities What is the probability that we get one “4” in 3 rolls?What is the probability that we get one “4” in 3 rolls? What is the probability that not all rolls are “4’s”What is the probability that not all rolls are “4’s” What is the probability that we get at least one “4”?What is the probability that we get at least one “4”?

Draw a Picture Roll a Die 4 4c4c 4 4c4c 4c4c c4c 4c4c 4c4c 4c4c Find the probability that we get one “4” in three rolls. Find the branches where this happens. Find the probabilities for each branch (1/6)(5/6)(5/6)=.1157 (5/6)(1/6)(5/6)=.1157 (5/6)(5/6)(1/6)=.1157 Then find the sum of the probabilities of these branches nd Roll 3 rd Roll

Draw a Picture Roll a Die 4 4c4c 4 4c4c 4c4c c4c 4c4c 4c4c 4c4c Find the probability that not ALL are “4’s” Find the probabilities for each branch OR notice that all branches except the branch where all are 4’s are checked….. So….we could find the P(not all) by using complements 1- P(all) 2 nd Roll 3 rd Roll 1-[(1/6)(1/6)(1/6)]=.9954 Find the branches where this happens

Draw a Picture Roll a Die 4 4c4c 4 4c4c 4c4c c4c 4c4c 4c4c 4c4c Find the probability that at least one is a “4” Find the probabilities for each branch OR notice that all branches except the branch where none are 4’s are checked….. So….we could find the P(at least one) by using complements 1- P(none) 2 nd Roll 3 rd Roll Find the branches where this happens 1-[(5/6)(5/6)(5/6)]=.4213

Additional Resources The Practice of Statistics—YMM The Practice of Statistics—YMM Pg Pg The Practice of Statistics—YMS The Practice of Statistics—YMS Pg Pg Against All Odds—Video #15 Against All Odds—Video #15 htmlhttp:// htmlhttp:// htmlhttp:// html