Probability
• Probability is the likelihood of a particular outcome occurring.
Law of Large Numbers
• As we repeat an experiment a large number of times, the ratio of the number of successes in the sample to the total number of trials will approach the probability of the event. • Example: proportion of heads should go to .5
10 coins: 100 coins: 1000 coins: 10000 coins: 7H 53 H 498 H 5002 H 3T 47 T 502 T 4998 T p = .7 p = .53 p = .498 p = .5002
• Example: probability of drawing a club from a deck of cards
• Example: in an urn full of 10 red marbles and 12 black marbles, the probability of drawing a black marble is
p= 12 12 = 10 + 12 22
Types of Probabilities
• Classical/Relative Frequency
Probability Definitions
• Sample Space
– S = list of all possible events that can occur – Example: faces of a die, S = {1,2,3,4,5,6}
• Subjective
– Likelihood of an event is based on your own personal judgment
• Event
– Subset of the sample space, one particular outcome of interest – Example: rolling an even number: E = {2,4,6} – Probability of rolling an even is 3 / 6 = .5
Example of a Venn Diagram
Probability Definitions
• A and B (intersection)
– A and B both occur together
• A or B (union)
– Either A occurs or B occurs or both occur
Probability (HW 5.3)
• Sample Space = {1,2,3,4,5,6,7,8} F = {6,7,8} G = {1,3,5,7}
• List outcomes in (F and G), compute P(F and G)
Probability Definitions
• A complement
– All possible events that are not in A
• Example:
– A = it’s snowing – Ac = it’s not snowing
• List outcomes in (F or G), compute P(F or G)
Probability (HW 5.3)
• Sample Space = {1,2,3,4,5,6,7,8} E = {2,3,4,5} • List the outcomes in E-complement.
Disjoint Sets
• Two events are disjoint if they contain no common elements • Disjoint sets cannot happen at the same time • Example: drawing a card that is both a club and a diamond; impossible
• Compute P(E-complement).
Disjoint Sets (HW 5.3)
• Sample Space = {1,2,3,4,5,6,7,8} E = {2,3,4,5} F = {6,7,8} G = {1,3,5,7} • Pick out the two disjoint sets.
Disjoint Sets (HW 5.3)
E = {2,3,4,5} F = {6,7,8} • Compute P(E and F) • Compute P(E or F)
Addition Rule and Complement
• General formula: P(A or B) = P(A) + P(B) – P(A and B)
Addition Rule Example
• We pick a card from a deck of 52 cards.
D = diamond drawn A = ace drawn P(D) = P(A) P(A and D) =
• If A and B are disjoint sets, then P(A or B) = P(A) + P(B) (special case)
• What is P(A or D) ? • What is P(Ac) ?
• Complement probability: P(Ac) = 1 – P(A)
Independence
• Two events A and B are independent if the following formula is true: P(A and B) = P(A) x P(B) • Whether A occurs has nothing to do with whether or not B occurs • A = your salary is more than $50,000 • B = you were born in July
Warning!
• Independent sets and disjoint sets are not the same! • Independent sets:
– P(A and B) = P(A) x P(B) – A and B can occur together, but neither affects the other
• Disjoint sets:
– P(A and B) = 0 – A and B cannot occur together
Independence (HW 5.1-5.2)
• 82% of people who visit an ice cream store buy ice cream. Here is a tree diagram for the sample space of 3 (independent) customers. • Y = Yes (purchase) • N = No (no purchase)
Independence (HW 5.1-5.2)
• 82% of customers buy ice cream. • Find the probability that all three people buy ice cream.
• Find the probability that nobody buys ice cream.
• Find the probability at least one person buys ice cream.
Disjoint vs Independent (HW 5.3)
Let P(A) = .35 and P(B) = .50. Compute P(A or B) if… (i) P(A and B) = .10
Conditional Probability
• The probability that one event occurs, given that another event has already occurred.
(ii) A and B are disjoint
• P (A | B) is read as “probability of A given B.” • If A and B are independent, then the conditional probability of A given B is just the probability of A. • Whether A occurs had nothing to do with whether B occurs, so “given B” should have no effect on A’s probability.
(iii) A and B are independent.
Conditional Probability (HW 5.3)
We have an urn full of marbles of the following colors: 5 green, 4 blue, 3 yellow, 2 red, 1 white (15 total). What’s the probability of drawing a primary color? (Red, yellow, or blue) Given that we drew a primary color, what’s the probability it was either red or yellow?
Checking for Independence
• If any of these formulas hold, A and B are independent:
1. P(A | B) = P(A) 2. P(B | A) = P(B) 3. P(A and B) = P(A) x P(B)
Assuming the marble drawn was not a primary color, what’s the probability it was not green?
•
If any of these do not hold, A and B are dependent.
Independence (HW 5.3)
Independence (HW 5.3)
• Find the probability the individual stayed in a cabin (C), given that they were at the beach (B).
• Find the probability an individual stayed in a cabin. • Are the events of staying in a cabin (C) and being at the beach (B) independent?
Random Variables
• Discrete
– A countable, whole-number of possible values. – The number of words in a magazine article – The number of clubs in a drawing of 10 cards
Discrete Probability Distribution
Two requirements:
1. Each individual p(x) is between 0 and 1, inclusive 2. All probabilities sum to 1 The mean of a discrete distribution:
MEAN = " x ! p ( x )
• Continuous
– An uncountable, infinite number of possible values, with decimals allowed. – The weight of an athlete – The time taken to complete a race
Also called average, or expected value
Mean of a Distribution (HW 6.1-6.2)
• Here’s a table for the probability of different category hurricanes.
Discrete Mean (HW 6.1-6.3)
• The Cash 4 lottery involves picking 4 numbers, each 0 to 9, and hence 10,000 combinations. If you pick the correct combination, you win $1200. • On a single bet, what’s the probability you win $1200?
Category 1 2 3 4 5
Probability 0.15 0.32 0.12 0.05
• Make a probability distribution for this problem. X is how much you win; find the probabilities with each outcome.
• Find the missing value and the mean/expected value of this data set.
• Is this discrete or continuous?
Discrete Mean (HW 6.1-6.3)
• Find the mean (expected winnings) on this lottery (in dollars). How many cents does this correspond to?
Binomial Distribution
• Requirements:
1. n independent trials (discrete) 2. Each trial takes one of 2 possible outcomes 3. Probability of “success” on each trial is p
•
Variables in study:
1. n = number of trials 2. p = probability of an individual success
Binomial or Not? (HW 6.1-6.3)
• A student is guessing on a multiple choice quiz, with 10 questions, each with 4 possible choices. X is the number of questions the student gets correct. • A 6-sided die is tossed. X is the number of dots on the top face.
More Binomial Characteristics
More Binomial Characteristics
SUMMARY
• If p .5
Skewed Left Binomial formulas:
mean = n ! p
standard deviation = n ! p ! (1 " p )
StatCrunch Commands
• Binomial Calculator
– Stat > Calculators > Binomial – n, p, x = ?, calculate
Normal Curve (Empirical Rule)
What’s the mean and s.d.?
• Normal Calculator
– Stat > Calculators > Normal – Mean, Standard Deviation – X >= or = 36.9% => 37%