STATS CHAPTER 5

What is the sum rule of probability

0 ≤ P ≤ 1 OR 0% ≤ P ≤ 100% → The sum of the probability of all simple events must be 1.

Based on the chart: 1) What is the probability that a student prefers English? 2) What is the probability that a student prefers Science? 3) What is the probability that a student is female? 4) What is the probability a student is female in Science? 5) What is the probability a student is female or prefers Science?

1) P(E) = (36)/(135) → .267 2) P(S) = (32)/(135) → .237 3) P(F) = (63)/(135) → 7/15 → .467 4) P(F∨s) = (13)/(135) 5) P(F or S) = (63)/(135)+(32)/(135)-(13)/(135)= (82)/(135)

What is conditional probability?

A conditional probability is a probability of an event B occurring knowing that event A has occurred. → P(A|B) = (P(A and B))/(P(B)) * P (B) is the GIVEN

What is a venn diagram in statistics?

A venn diagram is a picture with the entire population enclosed in a rectangular region with ovals that contain subsets of the population. Some ovals may overlap. → Rectangle = Entire Pop → Ovals = Subsets of Pop

If the two circles overlap on a venn diagram, what does the intersecting area of the diagram represent?

Any element common to event A AND event B is in the the intersecting area of the diagram.

What is the formula for how many outcomes you will have?

Choices∧# of spaces ° Three flips: 2∧3 = 8 outcomes

What are compound events?

Compound events are when you have more than one event (outcome of interest). → Includes union of events

What is relative frequency probability?

Empirical probability is relative frequency probability. → P(A) = (n(Obs. A))/total ° P(A) = (Number of observed outcomes)/(Total number of "tries")

What is the addition rule if two events are NOT mutually exclusive?

If two events are NOT mutually exclusive then P( A or B) = P(A) + P(B) - P(A and B) * The subtraction ensures we don't double count

What is the addition rule if two events ARE mutually exclusive?

If two events are mutually exclusive then P( A or B) = P(A) + P(B) * Note P(A and B) = 0 so first formula works in both cases!

What is the law of large numbers?

In the long run, as the sample size increases and increases, the relative frequencies of the outcomes get closer to the theoretical probability values.

What is a dependent event?

Occurrence of an event affects the probabilities of the events that occur after. → When you pull out a card and keep it out and then select. ° Keywords "in succession" "Without replacement"

What does it mean for an event to be independent?

Occurrence of one event doesn't effect the probabilities of events that occur after. → When I choose one card and put it back there is still a one out of 52 chance of it being there. ° Keywords: "With replacement"

What is the event?

Outcome of interest, or a subset of your sample space that is denoted with a capital letter. ° In a sample space if you are looking for a boy that is denoted as "B"

What is the probability of rolling a 2 on a fair die?

P(2) = (1)/(6) chance

What is the multiplication rule for independent events?

P(A and B) = P(A) × P(B) → We use the multiplication rule only when we want to know the probability of two events occurring together

What is the multiplication rule for dependent events?

P(A and B) = P(A) × P(B|A) → The P(B|A) is the probability of event B occurring given event A already occurred

What is the probability of landing on tails after flipping a coin once?

P(T) = (1)/(2) chance ° This is classical probability based on sample space.

What is the relative frequency probability of if when a coin is flipped 100 times, 32 of the outcomes are tails?

P(T)= (32)/(100) → P(T) = .32 ° The probability that you will get a tails after flipping a coin 100 times is 32%

What is the sample space of rolling a fair six-sided die?

S = {1,2,3,4,5,6}

What is classical probability?

The equal likelihood outcome is classical probability based on a sample space. → P(A) = (N(A))/(N(S)) ° P(A) = (Number of times event A occurs)/(Total number of outcomes in sample space)

What is probability?

The numerical measure between 0 and 1 that describes the likelihood an event will occur.

If the probability of getting cancer is P(C) = .23, what is the complement?

The probability of NOT getting cancer is → P(C') = 1 - P(C) → P(C') = 1 - .23 → .77 = complement

What is the complement?

The probability of an event NOT occurring. → The complement is equal to the 1 minus the probability the event occurs. P(A∧c) = P(A') → P(A') = 1 - P(A)

What does the following mean: P(A) → P of A

The probability that event A occurs.

What is a sample space?

The set of all possible outcomes of a chance experiment.

What does it mean to be mutually exclusive?

Two events are mutually exclusive if the events can't happen at the same time. → Ex) When testing for a disease you are either positive or negative, you can't be both.

What does a venn diagram of a complement look like?

U = A∧c

What is a union of events?

When the keyword is "OR." → P (A or B) Probability of event A occurring OR event B occurring OR both → P(A or B) = P(A U B)

What is an intersection of events?

When the keyword is "and" → P(A and B)= Probability of event A and B occur → BOTH HAVE TO HAPPEN IN ORDER TO BE TRUE

Suppose there are 100 total people in a study. Fifty have disease A, thirty have disease B, and 8 have both diseases. a. What is P(Bc) b. What is P(A or B) c. What is P(A|B) d. What is P(B|A)

a. 1 - 30/100 = 70/100 b. (50/100) + (30/100) - 8/100 = 72/100 c. 8/30 *30 is the given of B d. 8/50 * 50 is the given of A

Given the contingency table: a. What is the probability a person randomly selected person will prefer math given the person is female? b. What is the probability a randomly selected person who is a male given he likes art?

a. 24/63 → 8/21 OR .38 b. 6/17

Label Mutually Exclusive or Not Mutually Exclusive a. Car battery works vs. Car Battery doesn't work b. Having a positive test result or having a negative test result c. Ordering a ham sandwich or soup for lunch.

a. Mutually exclusive b. Mutually exclusive c. Not mutually exclusive

Suppose there are 100 total people in a study. Fifty have disease A , thirty have disease B and 15 have both diseases. Show that these events are independent. a. What percent of the population have disease A? b. What percent with disease B have disease A? Compare to the previous answer.

a. P(A) = 50/100 → 50% b. P(A|B) = 15/30 → 50% BECAUSE 50% AND 50% ARE EQUAL TO ONE ANOTHER THESE ARE INDEPENDENT BECAUSE P(A|B) = P(A)

Example 2: Suppose P(A) = 0.3, P(B) = 0.4 and P(A and B) = 0.12, are A and B independent or dependent?

→ .12 = (.3)(.4) → .12 = .12 THEREFORE, BECAUSE THE PRODUCT IS EQUAL THIS IS INDEPENDENT

Example 1: Suppose P(A) = 0.3, P(B) = 0.4 and P(A and B) = 0.16, are A and B independent or dependent?

→ 0.16 = (.3)(.4) → 0.16 ≠ 0.12 THEREFORE, BECAUSE THE PRODUCT IS NOT EQUAL THIS IS DEPENDENT

What is classical probability based on? What is relative frequency probability based on?

→ Classical probability is based on sample space → Relative frequency probability is based on a study/experiment

What is the sample space of flipping a coin once? Of three times?

→ One flip: S = {H,T} → Three flips: S = {HHH,THH,HTH,HHT,TTH,THT,HTT,TTT}

Suppose there four different colored marbles (green, red, blue and yellow) in an urn. ° The probability of choosing a green marble is 0.18. ° The probability of choosing a red marble is 0.26. °The probability of choosing a blue marble is 0.36 What is the probability of choosing a yellow marble?

→ P (Y) = 1 - [P(G) + P(R) + P(B)] → P (Y) = [.18 + .26+ .36] → P (Y) = 1 - .8 = .2 → Probability of choosing a yellow marble is .2

What is the probability of the first card out of a deck being a queen and the second card out of the deck being a king without replacement? → 52 cards in a deck, 4 queens, 4 kings

→ P(A and B) = P(A) × P(B|A) → (4/52) * (4/51) = → .006 or 4/663

If P(A) is close to zero what does that mean? If P(A) is close to one what does that mean?

→ P(A) close to zero means the event is unlikely (if it IS zero it NEVER happens) → P(A) close to 1 means the event is highly likely (if it IS one it ALWAYS happens)

How do you test whether two events are independent?

→ P(A|B) = P(A) → P(B|A) = P(B) or → P(A and B) = P(A)×P(B)

What is the probability of pulling out a King given that it is black? → 26 black cards, 4 kings

→ P(K|B) = 2/26 → 1/13 ° There are only 2 black kings!

What is the probability of pulling out a red card given that it is an 8? → 26 red cards, 4 eights

→ P(R|8) = 2/4 → 1/2 ° Of the 4 8s, 2 are red.

What is the probability of the first and second card out of a deck being red with replacement? → 52 cards in a deck, 26 red, 26 black

→ P(R₁ & R₂) = P(R₁) × P(R₂) → (26/52) × (26/52) = → .25 OR 1/4

What is the probability of the first and second card out of a deck being red without replacement? → 52 cards in a deck, 26 red, 26 black

→ P(R₁ and R₂) = P(R₁) × P(R₂|R₁) → (26/52) × (25/51) = → .245 OR 25/102

What is the sample space of a single birth? Of two births?

→ Single birth: S = {B,G} → Two births: {BB,BG,GB,GG}