PROBABILITY NMH UPWARD BOUND SUMMER ACADEMY STATISTICS HUNTER
PROBABILITY NMH UPWARD BOUND SUMMER ACADEMY STATISTICS HUNTER CHARLES REARDON
What is probability? v How likely is it that A will happen? v denoted by P and events are capitalicized letters v P(A) is read as “The probability of A happening” v. Some vocabulary v Event: what may or may not take place; v Can be simple or complex v. Sample space: all possible outcomes of an event
Calculating simple probabilities •
Some probability rules •
Your Turn v What is the probability of rolling a 6 with one die? v There are 10 rooms in Cutler basement. Your friend is in one of the rooms. If you choose one room at random, what is the probability you will find the one containing your friend? v In 2003, 389 of the 281, 421, 906 people in the United States were struck by lighting. Assuming the probability doesn’t change from year to year, what is the probability that you get struck by lightning? v Among 85 women who used various store-bought pregnancy tests, 5 tests yielded the wrong conclusion. What is the probability of a pregnancy test giving the right conclusion?
Addition Rule v P(A or B) = P(event A occurs or event B occurs or they both occur) v The formula for the addition rule is: v P(A or B) = P(A) + P(B) – P(A and B) v. Count every occurrence of A, then every occurrence of B, then subtract every occurrence of A and B together v. We want to avoid counting any one occurrence of an event twice. That will mess up our probability calculation!
Example • P(A): green pod P(A and B): Green pod and white flowers You can’t count them twice!! P(B): white flowers
Your Turn v If one of these 14 plants were to be chosen at random, what is the probability of picking one with a yellow pod or purple flowers?
Disjoint events v Definition: two events, A and B, are disjoint, or mutually exclusive, if they cannot both occur together. v Event A: at 12: 15 pm today, I am in the Upper Mod. v Event B: at 12: 15 pm today, I am in the RAC. v. These two events are disjoint because they cannot both be possible at the same time! v. Event A: Randomly selecting a female college student. v Event B: Randomly selecting a Hindu college student. v. These two events are not disjoint because they can occur together. A college student can be both female and Hindu!
Complement formula •
Crosstabulations v Crosstabulations, called crosstabs for short, are tables that show frequencies for two variables that are not disjoint. v. Called contingency tables by some textbooks and Pivot Tables in Excel 2007 on v. They are the easiest way to think about the addition rule (and the Multiplication Rule too!) v. They make probability calculations easier v. They should always include boxes for each variable intersection (think Punnett squares) and also boxes for your totals. Sometimes books will leave the totals off— add them yourself to make life easier! v. Sometimes this same data might show up in a pie chart. Move it to a crosstabulation to make life easier!
Example: Pg. 138 #9 -12 (from your homework) The following crosstabulation summarizes results from the Titanic disaster. Men Women Boys Girls Men Women Boys Survived 332 318 29 27 Survived 332 318 29 Died 1360 104 35 18 Died 1360 104 35 TOTAL 1692 422 64 45 TOTAL Girls 706 27 1517 18 n = 2223 If one of the Titanic passengers is randomly chosen, find the probability of getting someone who died or was a man. If one of the Titanic passengers is randomly chosen, find the probability of getting someone who survived or was a girl. If one of the Titanic passengers is randomly chosen, find the probability of getting someone who died. If one of the Titanic passengers is randomly chosen, find the probability of getting a boy.
Multiplication Rule v Basic Multiplication Rule: v. P(A and B) = P(A) • P(B) v. Read: The probability of event A and B equals the probability of event A times the probability of event B. v Think this through. v. Before with the addition rule, we were interested in A or B or both happening. Here, we are interested only in A and B happening together. Think about the football-shaped area in the middle of a Venn Diagram. v. As more events happen, the combined (and) probabilities tend to decrease.
Simple Examples •
The catch… v The Basic Multiplication Rule only works if the events are independent. v. Independent Events are two events that happen in such a way that one event does not affect how the other event happens. v. Ex: “My car starts this morning” and “My boss gets to work on time” are independent events v. Ex: “My car starts this morning” and “I get to work on time” are not independent events. v The probability of a person getting to work on time will change if it is given that their car doesn’t start. v. We’ll come back to this idea momentarily.
So what if they’re not independent? v You need the Formal Multiplication Rule. v. P(A and B) = P(A) • P(A|B) v. Read: “The probability of A and B is equal to the probability of A times the probability of A given B. v. If A and B are independent events, P(A|B) will be equal to P(B). v Ex: In the P(heads and 6) problem from earlier, that you got heads before you rolled the did not change the probability of rolling a 6 on the die. v Replacement v. This is the most common place you will see dependent events in probability outside of crosstabs v. Ex: If you draw a marble out of a bag and it is not replaced, the denominator of the fraction in the next probability calculation is reduced by one
Example •
Conditional Probability v The easiest way to think about this is in terms of crosstabulations v. If you know something happened, what is the probability of this other thing happening v. There’s a formula: v. You don’t need it! Just use the intuitive approach to conditional probability v. Assume that event A has occurred, and working under that assumption, calculating the probability that B will occur.
Example Men Women Boys Girls TOTAL Survived 332 318 29 27 706 Died 1360 104 35 18 1517 TOTAL 1692 422 64 45 n = 2223 1. Given that the passenger died, what is the probability that passenger was a woman? 2. Given that the passenger was a boy, what is the probability that he survived? 3. Given that the passenger was a child, what is the probability that child survived?
One last word v Significant figures! v You learned this in Chemistry…and in Biology…and in 8 th grade… v. When reporting probabilities, this is really important because it allows your data to have consistency v. Three numbers, not including leading zeroes, after the decimal point. v. Ex: Your calculator pumped out. 000543887465. What should you write down? v. The AP Test will dock points for that! (and so will I…)
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