STA215H5 Study Guide - Final Guide: Prior Probability, Test Statistic, Central Limit Theorem
CHAPTER 12 – PROBABILITY SAMPLING
1, 3, 11, 13, 15, 17, 19, 21, 23
Flip a coin. Can you predict the outcome? It’s hard to guess the outcome based on just one ip because the
outcome is random. If it is a fair coin though, you can predict the proportion of heads you’re likely to see in the
long-term. To do this, we need to talk about the probability of different outcomes.
PROBABILITY
= Numerical measure of the likelihood that an event will occur; number expressed between 0 and 1
1 → certainty
EX: Probability of a coin toss resulting in either heads or tails is 1 because there are no other options
0.5 → equal odds of occurring and not occurring
EX: Probability of a coin toss resulting in heads is 0.5 and tails is also 0.5
0 → impossibility
EX: Probability that the coin will land flat without either side is up is 0 because heads or tails must be up
Sample space (S): set of all possible outcomes of a random phenomenon
Trial: each occasion when we observe a random phenomenon
Outcome: the value of the random phenomenon at each trial
Event: outcome or set of outcomes of a random phenomenon; event is a subset of the sample space
EX: FAIR DIE
Rolling a fair die
Possible outcomes are: 1, 2, 3, 4, 5, 6
Probability of rolling a 4 is 1/6
EX: ORDER
In each of the following situations, describe the sample space S for the random phenomenon
a) A basketball player shoots three free throws. You record the sequence of hits and misses
The player could either get a hit (H) or miss (M)
S = {(HHH), (HHM), (MHH), (HMH), (MHM), (MMH), (HMM), (MMM)}
b) A basketball player shoots three free throws. You record the number of baskets she makes.
The player could either get 0, 1, 2, or 3 baskets
S = {0, 1, 2, 3}
COMBINATIONS
= Finding the number of combinations from the given population without looking at order
EX: GROUPS
How many groups of three individuals can we make from a group of 5
A, B, C, D, E
S = {(ABC), (ABD), (ABE), (BCD), (BCE), (CDE), (BDE), (ACD), (ACE), (ADE)} = 10 outcomes
N = 5
n = 3
PERMUTATIONS
= These deal with ordering; essentially, they are ordered combinations
= Combination x order
EX: GROUPS AND ORDER
How many different committees consisting of a president and a vice president can be selected from a group of
ve individuals?
= Combinations are only used for groups, not for order where you have to distinguish between the two roles
= The way we organize the groups matter
A, B, C, D, E
S = {(AB)(AC)(AD)(AE)(BC)(BD)(BE)(CD)(CE)(DE)} = 10 outcomes
S = {(BA)(CA)(DA)(EA)(CB)(DB)(EB)(DC)(EC)(ED)} = all 20 outcomes
EX: USING FORMULAS
A, B, C, D, E, F, G, H
N = 8
n = 3
How many ways can three books be selected from a group of eight books?
56
How many permutations of three books can be selected from a group of eight books?
336
EX: POWERBALL
a) Compute the number of ways the rst ve numbers can be selected
55 chooses 5 = 3,478,761
42 chooses 1 = 42
b. What is the probability of winning a prize of $200,000 by matching the numbers on the ve white
balls?
1/3,478,761 = 0.000000287
c) What is the probability of winning the Powerball jackpot?
Our probability of winning is 1/146,107,962 = 0.000000007
EX: FAIR DIE
Consider the experiment of rolling a fair pair of dice. Suppose that we are interested in the sum of the face
values showing on the dice
a) How many sample points are possible?
6 possibilities on first die and 6 on the other so 6 x 6 = 36
b) List the sample points
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
c) What is the probability of obtaining a value of 7?
6/36 = 1/6
d) What is the probability of obtaining a value of 9 or greater?
10/36
e) Because each roll has six possible even values (2,4,6,8,10 and 12) and only ve possible odd values
(3,5,7,9, and 11), the dice should show even values more often than odd values. Do you agree with this
statement? Explain.
Even though a die has 6 possible even values and 5 odd values, the probability of receiving even value differs
based on the 36 possibilities.
PROBABILITY RULES
(1) The probability P(A) of any event A satises 0 ≤ P(A) ≤ 1
(2) If S is the sample space in a probability model, then P(S) = 1
(3) For any event A, P(A does not occur) = 1 - P(A)
(4) Two events A and B are disjoint or mutually exclusive if they have no outcomes in common and therefore,
can never occur simultaneously
- If A and B are disjoint, P(A or B or both) = P(A) + P(B) → ADDITION RULE FOR DISJOINT
- If A and B are disjoint, P(A and B) = empty/0
GENERAL ADDITION RULE
= For any two events A and B, not mutually exclusive
P(A or B) = P(A)+P(B) - P(A and B)
Which we will write as: P(A∪B) = P(A) + P(B)−P(A∩B)
∪ = union
∩ = intersection
EX: CANDY
Suppose you draw a candy at random from a bag that has candies of seven different colours: blue, green,
purple, red, yellow, white, and orange.
COLOUR
Blue
Green
Purple
Red
Yellow
White
Orange
PROBABILITY
0.2
0.3
0.1
0.1
0.1
0.1
?
a) What must be the probability of drawing an orange candy?
0.1
b) What is the probability that you do not draw a green candy?
0.7
c) What is the probability that the candy you draw is either yellow, orange, or red?
0.3
EX: SMOKE/LUNG
Determine the probability that a man smokes or has lung disease
0.34
Marginal variable
CHAPTER 13 – VARIABLES AND BINOMIALS
1, 3, 7, 19, 21, 25, 27
CONDITIONAL PROBABILITY
EX: Is the probability of a car accident the same for all age groups? Insurance companies in particular, want to
know this so they can set their rates
EX: Is everyone equally likely in this room to get the u? Medical researchers use factors like age, sex,
lifestyle, and family history to estimate different u probabilities for different individuals
= Probability that takes into account a given condition
= Probability of an event, say event A, given that another, say event B, has already occurred
P(A|B) “A given B”
Document Summary
It"s hard to guess the outcome based on just one ip because the outcome is random. If it is a fair coin though, you can predict the proportion of heads you"re likely to see in the long-term. To do this, we need to talk about the probability of di erent outcomes. = numerical measure of the likelihood that an event will occur; number expressed between 0 and 1. Ex: probability of a coin toss resulting in either heads or tails is 1 because there are no other options. 0. 5 equal odds of occurring and not occurring. Ex: probability of a coin toss resulting in heads is 0. 5 and tails is also 0. 5. Ex: probability that the coin will land flat without either side is up is 0 because heads or tails must be up. Sample space (s): set of all possible outcomes of a random phenomenon. Trial: each occasion when we observe a random phenomenon.
