Chapter 14 From Randomness to Probability Randomness Probability
Chapter 14 From Randomness to Probability
Randomness & Probability Models: Behavior is random if, while individual outcomes are uncertain, for a large number of repetitions, outcomes are regularly distributed. A situation in which we know what outcomes could happen, but we don’t know which particular outcomes will happen is called a random phenomenon.
Example: If I roll a die once, I can’t predict with any certainty what number it will land on, but if I roll sixty times, I can expect it to land on 1 ten times, 2 ten times, 3 ten times, etc.
The probability of an outcome is the proportion of times the outcome would occur for a large number of repetitions (in the long run). Example: The probability of a die landing on 4 is the proportion of times a die lands on 4 for a large number of repetitions.
The set of all possible outcomes of an event is the sample space of the event. Example: For the event “roll a die and observe what number it lands on” the sample space contains all possible numbers the die could land on. S = {1, 2, 3, 4, 5, 6 }
An event is an outcome (or a set of outcomes) from a sample space. Example 1: When flipping three coins, an event may be getting the result THT. In this case, the event is one outcome from the sample space. Example 2: When flipping three coins, an event may be getting two tails. In this case, the event is a set of outcomes (TTH, THT, HTT) from the sample space.
An event is usually denoted by a capital letter. For example, call getting two tails event A. The probability of event A is denoted P(A).
Probability Rules: The probability of any event is between 0 and 1. A probability of 0 indicates the event will never occur, and a probability of 1 indicates the event will always occur. If S is the sample space, then P(S) = 1, since some outcome in the sample space is guaranteed to occur.
The probability that event A does not occur is one minus the probability that A does occur. That A will not occur is called the complement of A and is denoted P(AC). 1 – P(A) Example: When flipping two coins, the probability of getting two heads is 0. 25. The probability of not getting two heads is 1 -. 25 =. 75.
The union of two or more events is the event that at least one of those events occurs. A B Addition Rule for the Union of Two Events:
The intersection of two or more events is the event that all of those events occur. A B Multiplication Rule for the Intersection of Two Events:
Example: In our class of 36 students, we found that 25 students like to listen to rap music, 22 students like to listen to alternative music, and 16 students like to listen to both rap and alternative music. Find the probabilities of the following events: A: a student likes to listen to rap B: a student likes to listen to alternative A or B: a student likes to listen to rap or alternative A and B: a student likes to listen to rap and alternative
Example: In our class of 36 students, we found that 25 students like to listen to rap music, 22 students like to listen to alternative music, and 16 students like to listen to both rap and alternative music. Describe the following events: A c: B c: Ac or Bc: Ac and Bc:
If events A and B are disjoint or mutually exclusive (they have no outcomes in common), then the probability that A or B occurs is the probability that A occurs plus the probability that B occurs. P(A U B) = P(A) + P(B)
Example: Let event A be rolling a die and landing on an even number, and event B be rolling a die and landing on an odd number. The outcomes for A are { 2, 4, 6 } and the outcomes for B are { 1, 3, 5 }. These events are disjoint because they have no outcomes in common. So the probability of A or B (landing on either an even or an odd number) equals the probability of A plus the probability of B.
Events A and B are independent if knowing that one occurs does not change the probability that the other occurs. Example: Roll a yellow die and a red die. Event A is the yellow die landing on an even number, and event B is the red die landing on an odd number. These two events are independent because the occurence of A does not change the probability of B.
If events A and B are independent then the probability of A and B equals the probability of A times the probability of B. P(A) X P(B)
Example: The probability that the yellow die lands on an even number and the red die lands on an odd number is:
If events A and B are independent, then their complements, Ac and Bc are also independent and Ac is independent of B.
The probability that event A occurs if we know for certain that event B will occur is called conditional probability. The conditional probability of A given B is denoted: P(A|B) If events A and B are independent then knowing that event B will occur does not change the probability of A so for independent events: P(A)
Example: When flipping a coin twice, what is the probability of getting heads on the second flip if the first flip was a head? Event A: getting heads on first flip Event B: getting heads on second flip Events A and B are independent since the outcome of the first flip does not change the probability of the second flip, so…
Are events A and B independent? Suppose you draw one card from a standard deck of cards. A = you draw an ace B = you draw a heart
Practice Problems 1) If A U B = S (sample space), P(A and Bc) =. 2, and P(Ac) =. 5, then P(A) = ____.
2) Given: P(A) =. 4, P(B) =. 25, and P(A and B) =. 1. What is P(A|B)?
3) If P(A) =. 24, P(B) =. 62, and A and B are independent, what is P(A or B)?
4) You play tennis with a friend, and from past experience, you believe that the outcome of each match is independent. For any given match you have a probability of. 6 of winning. What is the probability that you win at least one of the next 4 matches?
5) Suppose that you have a torn tendon and are facing surgery to repair it. The surgeon explains the risks to you. Infection occurs in 3% of such operations, the repair fails in 14%, and both infection and failure occur together in 1%. What percent of these operations succeed and are free from infection?
6) Suppose that for a group of consumers, the probability of eating pretzels is. 75 and the probability of drinking coke is. 65. Further suppose that the probability of eating pretzels and drinking coke is. 55. Determine if these two events are independent.
p. 339: # 20 Stats Project In a large Introductory Statistics lecture hall, the professor reports that 55% of the students enrolled have never taken a Calculus course, 32% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on the project for the course. What is the probability that the first groupmate you meet has studied a) two or more semesters of calculus? b) some Calculus? c) no more than one semester of Calculus?
p. 339: # 22 Another Project You are assigned to be part of a group of three students from the Intro Stats class described in Exercise 20. What is the probability that of your other two groupmates, a) neither has studied Calculus? b) both have studied at least one semester of Calculus? c) at least one has had more than one semester of Calculus?
p. 340: # 26 Failing fathers? A Pew Research poll in 2007 asked 2020 U. S. adults whether fathers today were doing as good a job of fathering as fathers of 20 -30 years ago. Here’s how they responded: Response Number Better 424 Same 566 Worse 950 No Opinion 80 TOTAL 2020 If we select a respondent at random from this sample of 2020 adults, what is the probability that a) the selected person responded “Worse? ” b) the person responded the “Same” or “Better? ”
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