P2010 Probability (Ch 5)
A. Definition of probability
Probability is a number attached to an event that has not yet occurred.
The probability of an event is a number between 0 and 1 inclusive.
This number represents the likelihood of occurrence of the event.
B. Examples of Events
The event "The stoplight will not turn red before I enter the intersection"
The event "It will snow enough to have school cancelled tomorrow".
The event "A coin tossed into the air lands with Head facing up".
The event "I change my name to Fred tomorrow".
C. Why do we want probability values, i.e., the actual numbers?
1. A probability value is a formalized way of looking into the future and making decisions based on the probability.
When our oldest son was 4, we found that he had a tumor on his neck.
The doctors, based on the available evidence, gave us the probability of his survival if we did not operate and the probabilities of various outcomes if we did operate. It was easier for us to compare the various alternatives when expressed as probabilities of success.
The consequences of a pre-emptive strike on ISIS oil production facilities have been expressed as probabilities – the probability of a failure, the probability of a Mideast uprising, the probability of ISIS backing down in its attempts to create a caliphate, the probability of a strike strengthening ISIS. The values are easily comparable.
2. Probability values can be easily related to other variables, such as costs.
Insurance companies determine the probability of persons of each age having a car accident and set rates based on those probability estimates – the higher the probability, the higher the rate. The probability -> cost relationship is easily used once the probability values have been obtained.
3. Probabilities are used to evaluate the results of experiments
An experiment is conducted comparing two pain relievers and the outcome is recorded.
The probability of that particular outcome is computed assuming that the pain relievers are equal.
If that probability is larger than .05 (usually), then the researcher concludes that the pain relievers are equal.
But if the probability is less than or equal to .05, then the researcher concludes that they’re not equal.
From a recent report on when you should take blood pressure medicine. Specifically, the odds of type 2 diabetes dropped 61 percent for people taking angiotensin receptor blockers at bedtime compared to morning. For those on ACE inhibitors at night, the odds went down 69 percent. People on beta blockers reduced their odds of the blood sugar disease by 65 percent when they took their medicine at night, the researchers reported.
D. Ways of computing or determining the probability of an event.
1. Subjective estimates.
“Will the light turn red before I get to the intersection. Hmm. There are no cars waiting at the light. That must mean that it hasn’t been green very long, so it’ll probably be green when I get there.”
2. The Relative Frequency or actuarial method, from the proportion of previous occurrences,
Number of previous occurrences
Probability estimate = ------
Number of previous opportunities to occur
Problems with this method
Not applicable to events for which we have no history (AIDS)
Accurate counts may not be available (AIDS again)
Applications of this method
Actuarial tables created by insurance companies
3. From knowledge of the sampling distribution of specific statistics, such as the sample mean.
We will use the normal distribution to determine probabilities of specific experimental outcomes.
Combination Events: Events defined as combinations of other events, say A and B .
1. The intersection event, “A and also B”
The intersection is the event that occurs when both A and also B occur.
This event is called the Intersection of A and B. The name can best be appreciated by considering A and B as two streets which cross.
Example: A = The first person past the door is a psych major.
B = The first person past the door is male.
Intersection event: The first person past the door is a psych major and a male.
2. The union event, “Either A or B or Both”.
The union is the event that occurs if either A occurs or B occurs or Both occur.
This event is called the Union of A and B. Think of union as in the union of matrimony. After the marriage, either the husband (A) or the wife (B) or both can represent the family.
Example: A = The first person past the door is a psych major.
B = The first person past the door is female.
Union event: The first person past the door is either a psych major or female or both.
Two important event relationships and a law identified with each.
1. Mutually exclusive events.
Two events, A and B, are mutually exclusive if the occurrence of either precludes the occurrence of the other.
Examples.
A: A coin lands with Head facing up.
B: On the same toss, the coin lands with Tail facing up.
A: 1st person past the door is 6’ tall or taller as he/she walks by.
B: 1st person past the door is less than 6’ tall as he/she walks by.
A: The IQ of a randomly selected person is greater than 130 when measured.
B: The IQ of the same randomly selected person is less than 70 when measured.
Visual representation of mutually exclusive events: (Venn diagram)
In the universe of possible event, there is no overlap between A and B. So if A occurs, B cannot occur and vice verse.
The additive law of probability for mutually exclusive events.
A Probability law identified with mutually exclusive events
If A and B are mutually exclusive, P(Either A or B) = P(A) + P(B).
Example: Suppose a single die is tossed.
A: Side facing up has exactly 1 dot.
B: Side facing up has either 5 or 6 dots.
Whimsical Example
A = First creature past the door is a psych major P = .4.
B = First creature past the door is a Martian P = 0
P(First creature past the door is either a psych major or a Martian) = .4 + 0 = .4.
2. Independent events.
Two events, A and B, are independent if the occurrence of either has no effect on the probability of occurrence of the other.
Examples of events are are probably independent.
A: It will rain in Tokyo tomorrow.
B: The Braves will win the first home game of next season.
A: Jerry Smith, a person who I met this morning on Market St. will vote Republican in 2014.
B: Samuel Johnson, a person who works in retail in New York, will vote Republican in 2014
Examples of events that are NOT independent
A: John Smith will vote Republican in the next election.
B: Mary Smith, John’s wife, will vote Republican in the next election
A. Jim Samuels, a person I met on Broad St. this AM, will vote for the Democratic Congressional candidate in the next election.
B: Jim Samuels, the same person will vote Republican Congressional candidate in the next election.
The multiplicative law of probability for independent events.
A probability law identified with independent events.
If A and B are independent, P(A and also B) = P(A)*P(B)
Example:
A: Experimenter A finds difference due to chance alone. P(A) = .05.
B: Experimenter B finds difference due to chance alone. P(B) = .05.
What’s the probability that A and also B will find a difference.
Suppose A and B are independent.
Suppose P(A) = .05 and P(B) = .05.
Then P(A and also B) = .05 * .05 = .0025.
Exercises . . .
1. A really good football team is so good that the probability of it winning any game is .9.
Assuming successive games are independent, what’s the probability of that team going undefeated in a 10 game season? This is an application of the multiplicative law to 10 events.
Game 1 Game 2 Game 3 Game 4 Game 5 Game 6 Game 7 Game 8 Game 9 Game 10
.9 .9 .9 .9 .9 .9 .9 .9 .9 .9
Answer is .910 = .3486.
2. How good would it have to be to have a 50-50 chance of winning all games?
.95? .9510 = .60 So somewhere between .9 and .95.
3. How good would it have to be to have a 90% chance of winning all games?
.99? .9910 = .90
Biderman’s P201 Handouts Topic 10: Probability - 1 9/24/2015