Transcript AP Statistics

AP STATISTICS
Section 6.2 Probability Models
Objective: To be able to understand and apply the rules for
probability.
Random: refers to the type of order that reveals itself after
a large number of trials.
Probability of any outcome of a random phenomenon is the
proportion of times the outcome would occur in a very long
series of repetitions.
Types of Probability:
1. Empirical: probability based on observation.
Ex. Hershey Kisses:
2. Theoretical: probability based on a mathematical model.
Ex. Calculate the probability of flipping 3 coins and getting
all head.
Sample Space: set of all possible outcomes of a random
phenomenon.
Outcome: one result of a situation involving uncertainty.
Event: any single outcome or collection of outcomes from
the sample space.
Methods for Finding the Total Number of Outcomes:
1. Tree Diagrams: useful method to list all outcomes in the
sample space. Best with a small number of outcomes.
Ex. Draw a tree diagram and list the sample space for the event
where one coin is flipped and one die is rolled.
2.
Multiplication Principle: If event 1 occurs M ways and event
2 occurs N ways then events 1 and 2 occur in succession
M*N ways.
Ex. Use the multiplication principle to determine the
number of outcomes in the sample space for when 5 dice
are rolled.
Sampling with replacement: when multiple items are being
selected, the previous item is replaced prior to the next
selection.
Sampling without replacement: then the item is NOT
replaced prior to the next selection.
Rules for Probability
Let A = any event;
1.
𝑃 𝐴 =
Let P(A) be read as “the probability of event A”
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝐴 𝑜𝑐𝑐𝑢𝑟𝑠
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
0≤𝑃 𝐴 ≤1
3. If P(A) = 0 then A can never occur.
4. If P(A) = 1 then A always occurs.
5.
𝑃 𝐴 = 1; the sum of all the outcomes in S equals 1.
6. Complement Rule: 𝐴𝑐 or 𝐴′ is read as “the complement
of A”
𝑃 𝐴𝑐 is read as “the probability that A does NOT occur”
𝑃 𝐴𝑐 = 1 − 𝑃(𝐴) or 𝑃 𝐴𝑐 + 𝑃 𝐴 = 1
Key words: not, at least, at most
2.
Ex. 1 Roll one die, find 𝑃 6𝑐
Ex. 2 Flip 5 coins, find P(at least 1 tail)
7. The General Addition Rule: (use when selecting one
item)
𝑃 𝐴 𝑜𝑟𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)
Ex. Roll one die, find 𝑃(< 3 𝑜𝑟 𝐸𝑣𝑒𝑛)
Ex. Roll one die, find 𝑃(< 3 𝑜𝑟 > 4)
Events A and B are disjoint if A and B have no elements in
common. (mutually exclusive)
𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 0
𝐴∩𝐵 =∅
Ex. Choose one card from a standard deck of cards. Find
𝑃 𝑅𝑒𝑑 𝑜𝑟 𝐾𝑖𝑛𝑔
𝑃 𝐷𝑖𝑎𝑚𝑜𝑛𝑑 𝑜𝑟 8
𝑃(𝐹𝑎𝑐𝑒 𝑐𝑎𝑟𝑑 𝑜𝑟 𝑏𝑙𝑎𝑐𝑘)
𝑃 𝑘𝑖𝑛𝑔 𝑎𝑛𝑑 𝑞𝑢𝑒𝑒𝑛
𝑃(𝐻𝑒𝑎𝑟𝑡 𝑜𝑟 𝑆𝑝𝑎𝑑𝑒)
𝑃 𝑃𝑖𝑐𝑡𝑢𝑟𝑒 𝑐𝑎𝑟𝑑 𝑜𝑟 10
8. Equally Likely Outcomes: If sample space S has k
equally likely outcomes and event A consists of one of
1
these outcomes, then 𝑃 𝐴 =
Ex.
𝑘
9. The Multiplication Rule: (use when more than one item
is being selected)
If events A and B are independent and A and B occur in
succession, the 𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃 𝐴 ∙ 𝑃 𝐵
Events A and B are said to be independent if the
occurrence of the first event does not change the
probability of the second event occurring.
Ex. TEST FOR INDEPENDENCE. Flip 2 coins, let A =
heads on 1st and B = heads on 2nd. Are A and B
independent?
Find 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
Find 𝑃(𝐴) ∙ 𝑃(𝐵)
Any events that involve “replacement” are independent and
events that involve “without replacement” are dependent.
Ex. Choose 2 cards with replacement from a standard
deck. Find
𝑃(𝐴𝑐𝑒 𝑎𝑛𝑑 𝐾𝑖𝑛𝑔)
𝑃(10 𝑎𝑛𝑑 𝐹𝑎𝑐𝑒 𝐶𝑎𝑟𝑑)
Repeat without replacement:
𝑃(𝐴𝑐𝑒 𝑎𝑛𝑑 𝐾𝑖𝑛𝑔)
𝑃(10 𝑎𝑛𝑑 𝐹𝑎𝑐𝑒 𝐶𝑎𝑟𝑑)
IF EVENTS ARE DISJOINT, THEN THEY CAN NOT BE
INDEPENDENT!!!!!
Ex.
Let A = earn an A in Statistics; P(A) = 0.30
Let B = earn a B in Statistics; P(B) = 0.40
Are events A and B disjoint?
Are events A and B independent?
Independence vs. Disjoint
Case 1) A and B are NOT disjoint and independent.
Suppose a family plans on having 2 children and the P(boy) = 0.5
Let A = first child is a boy.
Let B = second child is a boy
Are A and B disjoint?
Are A and B independent? (check mathematically)
Case 2) A and B are NOT disjoint and dependent. (Use a Venn
Diagram for Ex)
Are A and B disjoint?
Are A and B independent? (check mathematically)
Case 3) A and B are disjoint and dependent.
Given P(A) = 0.2 , P(B) = 0.3 and P(A and B) = 0
Are A and B independent? (check mathematically)
(Also refer to example for grade in class)
Case 4) A and B are disjoint and independent.
IMPOSSIBLE
Ex. Given the following table of information regarding meal plan and
number of days at a university:
Day/Meal
Plan A
Plan B
2
0.15
0.20
5
0.20
0.25
7
0.05
0.15
Total:
Total:
A student is chosen at random from this university, find
P(plan A)
P(5 days)
P(plan B and 2 days)
P(plan B or 2 days)
Are days and meal plan independent? (verify mathematically)