Probability 1

A deck of playing cards has 52 cards, of which 12 are face cards. If you shuffle the deck well and turn over the top 3 cards, one after the other, what's the probability that all 3 are face cards?

0.010 Since you turn one after another, you are NOT replacing. (12/52)(11/51)(10/50)

A soft drink dispenser can be adjusted to deliver any fixed number of ounces of soft drink. If the machine is operating with a standard deviation in delivery equal to 0.3 ounces, what should be the mean setting so that a 12-ounce cup will overflow less then 1% of the time? Assume a normal distribution for ounces delivered.

11.30 ounces x = 12 stDev = 0.3 Mean = ?????

When two fair dice are rolled, what is the probability of getting a sum of 7, given that the first die rolled is an odd number?

6/36 = 1/6 If you want a sum of 7, one dice has to be odd!

Define Complement

If P(rain) = 0.30, then the P(not rain) = 0.70 Subtract from 1!

What is another way to think of independence using what we know about scatterplots (correlation)?

If two things have no correlation (r = 0), then they are independent. They don't affect each other.

When can a venn diagram be useful?

When two events overlap! When they are NOT disjoint.

What is conditional probability?

Given that A happened first, what are the chances of B also happening.

When can a probability tree diagram be useful?

Helps show the entire sample space and shows probability of all events happening. Most conditional probabilities can be solve with a tree diagram.

A spinner has 8 equal sections: three are numbered 1, one section is numbered 2, and the other four sections are numbered 3. The spinner has spun twice! What is the probability that the sum of those spins is 5?

If you spin 2 first, then 3...(1/8)(1/2) = 1/16 P(3) = (4/8) = (1/2) P(2) = (1/8) If you spin 3 first, then 2...(1/2)(1/8) = 1/16 So the chances you get sum of 5 is the first OR the second happening: (1/16) + (1/16) = 1/8

Are independent events mutually exclusive?

NEVER! The events below are disjoint. (You can't make an A and B in a class) A = you make an A in this class B = you make a B in this class. I say, "You didn't make an A." Then you know it's more likely you made a B. Since my information changed the probability of making a B, it is DEPENDENT.

What is the probability that given a heart, you draw an even-numbered card?

Option 1: 1) There are 13 hearts. 2) There are 5 even cards that are hearts...so 5/13 Option 2: 1) There are 13/52 hearts. 2) There are 5/52 even hearts...so (5/52)/(13/52) = 5/13 same answer!

How can you test if two events are independent?

P(A and B) = P(A) * P(B) P(A, given B) = P(A) [Knowing B didn't affect the probability of A]

Define Independence

When two events are said to be independent of each other, what this means is that the probability that one event occurs in no way affects the probability of the other event occurring. An example of two independent events is as follows; say you rolled a die and flipped a coin

James plays Mahir in tennis. From James' past experience, he believes he has a 60% chance of winning. Assume the outcome of each match is independent. What are the chances of Mahir winning all 3 games against James?

(0.4)(0.4)(0.4) = 0.064 Since they are independent, we can just multiply all three. One games doesn't affect the other. No conditional needed!

A plumbing contractor obtains 60% of her boiler circulators from a company whose defect rate is 0.005, and the rest from a company whose defect rate is 0.010. What proportion of the circulators can be expected to be defective? If a circulator is defective, what is the probability that it came from the first company?

.0070, .429

Suppose 4% of the population have a certain disease. A laboratory blood test gives a positive reading for 95% of people who have the disease and for 5% of people who do not have the disease. What is the probability of testing positive? If a person tests positive, what is the probability the person has the disease?

.086, .442

You toss a coin 5 times. Assume the coin is fair and that each outcome is independent. What are the chances of getting at least one head?

0.97 This is opposite of getting no heads! 1 - (0.5)^5

Koen and Devonte are scheduled to take their stats exam. The probability that Koen passes is 0.40. The probability that Devonte passes 0.60. If the two events are independent, which statement is correct? A) The probability that exactly one passes is 0.0576 B) The probability that exactly one fails is 0.52 C) The probability they both pass is 0.20 D) The probability they both pass is 1.0 E) The probability they both fail is 1.0.

B P(D pass AND K fails) = (0.60)(0.60) P(D fails AND K passes) = (0.4)(0.4) Since either the first OR the second can happen, add them up!

If two events are independent, what is the probability that both events occur? A) zero B) one C) the sum of their probabilities D) the product of their probabilities E) the difference in their probabilities

D

For events A and B related to the same chance process, which of the following statements is true? A) If A and B are mutually exclusive, then they must be independent. B) If A and B are independent, then they must be mutually exclusive. C) If A and B are not mutually exclusive, then they must be independent. D) If A and B are not independent, then they must be mutually exclusive. E) If A and B are independent, then they cannot be mutually exclusive.

E

Define Mutually Exclusive

Events are disjoint. Events share no outcomes in common! P(A and B) = 0

In 2000, it was the election between Bush and Gore. Suppose 2/3 of the people in the city supported Bush, but 5/9 of the people in the suburbs supported Gore. 60% of the people live in the city, where 40% live in suburbs. If you select a random Bush supporter, what is the probability they live in the city?

P(city, given Bush) = 0.692 P(city and Bush) = (0.6)(2/3) = 0.4 P(Bush) = (0.6)(2/3) + (.4)(4/9) = .577778 (0.4)/(0.577778) = 0.692 Try using the tree diagram! Start with city and suburbs, then break it up whether the support Bush or Gore.

What is the General Addition Rule?

Remember: You MUST subtract what they share so we don't overlap.

Define Probability Model

Shows all the outcomes in the sample space with the probabilities of each.

Define Probabiliy

The LONG RUN relative frequency of an event. Example: Flipping a fair coin; there is a 50% chance of getting a head in the LONG RUN