Probability Basics

Probability Basics Chapter 14 Basic concepts and vocabulary Open doc “45- class practice “ on the N-spire

Pg 1.2 Things we will cover. Probability Law of Large Numbers Sample Space Probability Assignment Rule Complement Rule Addition Rule Multiplication Rule

Pg 1.3 Probability Basics The probability of an event is a number between 0 and 1 It reports the likelihood of an event’s occurrence.

Day Light % Red Lights Probability 1 Red (1/1) 100 1.0 2 Green (1/2) 50 0.5 3 Red (2/3) 66.7 0.667 4 Red (3/4) 75 0.75 5 Green (3/5) 60 0.6 6 Green (3/6) 50 0.5 Observational Probability Determining the probability by what you see happens: As you increase the number of trials the Observational Probability gets closer to the actual probability.

100 100 90 80 70 60 50 40 30 Percent of Red 20 10 0 0 1 2 3 4 5 6 6 # of Outcomes

100 100 90 80 70 60 50 40 30 Percent of Red 20 10 0 0 1 2 3 4 5 6 6 # of Outcomes The graph, over time, settles down to 35% or we say P(red) = 0.35 Called “Law of Large Numbers”

Law of Large Numbers The long-run relative frequency of repeated events gets closer and closer to the true relative frequency as the number of trials increases. The idea that something is due to happen just because it hasn’t happened for a while is a false conception – there is no “law of averages”.Just because something hasn’t happened for a while does not change the probability that it will occur. The P(rolling a 6) is always going to be 1/6 regardless of how long its been since you rolled a 6.

Probability Assignment Rule“the duh rule” 0 ≤ P ≤ 1 and ∑P = 1

Complement Rule Suppose that P(on time) = P(T) = 0.85 What’s P(not on time) ? P(not on time) = P(TC) = 1 – P(T) = 0.15 Because P(T) + P(TC) = 1

Must be Disjoint Addition Rule The 2 events cannot happen at the same time. One excludes the other from happening! Suppose that S stands for a Sophomore and J stands for Junior. The probability of a randomly selected student is a Sophomore is 0.4 and the probability of a student being a Junior is 0.32. The P(S or J) = P(S) + P(J) = 0.4 + 0.32 = 0.72

Must be Independent Multiplication Rule The 2 events cannot affect the probability of each other. One does not effect the other happening. Suppose that S stands for a green light on the first day and J stands for a green light the 2nd day. The probability of getting a green light is 0.75. The probability of hitting green lights on 2 consecutive days is… The P(S and J) = P(S) x P(J) = 0.75 x 0.75 = 0.5625

Class practice • Make sure you are logged into the calculator and open 45- class practice.

Pg 2.1 In class practice: • You roll a fair die three times. What is the probability that . . . • you roll all 5’s? • are the events disjoint, independent, or neither

Pg 2.2 In class practice: • You roll a fair die three times. What is the probability that . . . 2) you roll all odd numbers?

Pg 2.3 In class practice: • You roll a fair die three times. What is the probability that . . . 3) none of your rolls gets a number divisible by 3?

Pg 2.4 In class practice: • You roll a fair die three times. What is the probability that . . . 4) you roll at least one 5?

Pg 2.5 In class practice: • You roll a fair die three times. What is the probability that . . . 5) The numbers you roll are not all 5’s?

Pg 3.1 to 3.4 • Census reports for a city indicate that 62% of the residents classify themselves as Christian, 12% as Jewish, and 16% as member's of other religions (Muslims, Buddhists, etc.). The remaining residents classify themselves as nonreligious. A polling organization seeking information about public opinions wants to be sure to talk with people holding a variety of religious views, and makes random phone calls. Among the first four people they call, what is the probability they reach . . . 6) all Christians? 7) no Jews? 8) at least one person who is nonreligious?

The probability of an event is its long term relative frequency found by . . . 1- Looking at many replications(repeats) of an event 2- Deducing it from equally likely events 3- Using some other information (the catch all)

Pg 4.1 to 4.3 A few years ago M&M’s decided to add a new color and conduct a global survey. In Japan they found that 38% chose pink, 36% teal and 16% chose purple and 10% had no preference Check to see if probabilities are legitimate – check to see if all are between 0 and 1 check the “duh” rule – all add to 1 (probability assignment rule) They don’t pass the “duh” rule so create a final category.

Pg 4.4 What’s the probability a random respondent preferred either pink or teal? The word “either” suggest the addition rule or P(A + B). Check if disjoint. P(pink or teal) = P(pink) + P(teal) = 0.38 + 0.36 = 0.74

Pg 4.5 What’s the probability that 2 random respondents both picked purple? The word “both” suggest the multiplication rule or P(A and B). Check for independence. P(both purple) = P(1st picks purple) x P(2nd picks purple) = 0.16 x 0.16 = 0.0256

Pg 4.6 What’s the probability that out of 3 random respondents at least 1 picked purple? The words “at least 1” suggest the complement rule or P(A) = 1 – P(Ac) Check for independence. P(not purple) = 1 – P(purple) = 1 – 0.16 = 0.84 P(none picking purple) = (0.84)3 = 0.5927 P(at least 1 purple) = 1 – P(none picked purple) = 1 – 0.5927 = 0.4073

Probability Basics