Transcript probability

SIMULATION
THE ACT OF IMITATING AN ACTUAL EVENT, CONDITION, OR
SITUATION.
WE OFTEN USE SIMULATIONS TO MODEL EVENTS THAT
ARE TOO LARGE OR IMPRACTICAL TO LITERALLY TEST.
FOR EXAMPLE, IF YOU WANTED TO KNOW THE
PROBABILITY OF HAVING A BOY OR GIRL YOU WOULD NOT
LITERALLY ATTEMPT THE EVENT. YOU WOULD RUN A
SIMULATION.
LET’S TRY IT.
PROCEDURE
1.Run a simulation to find the gender
of each child in 10 families.
2.Each family will have 3 children.
3.Flip a coin to determine the sex of
each child.
4.Heads = Male
5.Tails = Female
6.Record your results in the table
Family
1
2
3
4
5
6
7
8
9
10
Child 1
Child 2
Child 3
ANALYSIS
• How many families had three male
children? Female children?
• How many of your families had the same
order of male and female children?
• How many different combinations of
offspring are possible in this simulation?
• Did anyone else in the class have exactly
the same simulation results as you?
• Why is the gender of each child an
independent event?
HOW CAN I DETERMINE IF THE PROBABILITY IS
INDEPENDENT OR DEPENDENT?
INDEPENDENT
DEPENDENT
• WHATEVER HAPPENS IN ONE EVENT HAS ABSOLUTELY • WHAT HAPPENS DURING THE SECOND EVENT DEPENDS
NOTHING TO DO WITH WHAT WILL HAPPEN NEXT
UPON WHAT HAPPENED BEFORE
INDEPENDENT EVENTS ARE INDEPENDENT
BECAUSE…
1. THE TWO EVENTS HAVE NOTHING TO DO WITH ONE ANOTHER
2.
3.
OR
YOU REPEAT THE SAME ACTIVITY, BUT YOU REPLACE THE
ITEM THAT WAS REMOVED.
OR
YOU REPEAT AN EVENT WITH AN ITEM WHOSE NUMBERS WILL
NOT CHANGE (SPINNERS / DICE / ETC.)
TEST YOUR KNOWLEDGE…
ARE THE FOLLOWING INDEPENDENT OR DEPENDENT EVENTS?
YOU TOSS TWO
DICE AND GET A 5
ON BOTH OF THEM
INDEPENDENT
YOU HAVE A BAG OF MARBLES: 4
WHITE, 5 BLACK, 3 BLUE, 6 PURPLE,
AND 10 GREEN. YOU PULL ONE
MARBLE OUT OF THE BAG, LOOK AT
THE COLOR AND PUT IT BACK IN THE
BAG. THEN, YOU CHOOSE ANOTHER
MARBLE.
INDEPENDENT
YOU PULL A QUEEN OF
DIAMONDS, THEN A 3 OF
SPADES, AND FINALLY A 10
OF HEARTS FROM A DECK
OF CARDS WITHOUT
PUTTING ANY BACK IN.
DEPENDENT
The probability of an event is a number between 0 and 1 that
indicates the likelihood the event will occur.
P=0
Event will not occur
P = 1/2
Event is equally likely
to occur / not occur
You can express a probability as a fraction,
a decimal, or a percent.
For example: 1 , 0.5, or 50%.
2
P=1
Event is certain to
occur
THEORETICAL VS. EXPERIMENTAL PROBABILITY
THEORETICAL
• IN THEORY, THE PROBABILITY OF AN
EVENT WILL HAPPEN AS A RESULT
OF THE NUMBER OF FAVORABLE
OUTCOMES DIVIDED BY THE TOTAL
NUMBER OF OUTCOMES
P (A) =
number of outcomes in A
total number of outcomes
EXPERIMENTAL
• ONE CONDUCTS A PHYSICAL
EXPERIMENT TO DETERMINE THE
PROBABILITY OF AN EVENT
OCCURRING
HOW DO YOU FIND THE PROBABILITY?
• THE PROBABILITY OF TWO
INDEPENDENT EVENTS, A AND
B, IS EQUAL TO THE
PROBABILITY OF EVENT A
TIMES THE PROBABILITY OF
EVENT B.
• THE PROBABILITY OF TWO
DEPENDENT EVENTS, A AND B,
IS EQUAL TO THE PROBABILITY
OF EVENT A TIMES THE
PROBABILITY OF EVENT B.
HOWEVER, THE PROBABILITY
OF EVENT B NOW DEPENDS
ON EVENT A.
P(A, B) = P(A)  P(B)
“AND” INTERSECTIONS (∩)
INDEPENDENT
• Remember, this means the occurrence of
one does not change the probability of
the other occurring.
• 𝑃 𝐴 𝑎𝑛𝑑 𝐵
= 𝑃 𝐴 ∙ 𝑃(𝐵)
DEPENDENT
• Remember, this means the occurrence of
one affects the probability of the other
occurring.
•𝑃𝐵𝐴
means the probability of B given
A has already occurred
• 𝑃 𝐴 𝑎𝑛𝑑 𝐵
=𝑃 𝐴 ∙𝑃 𝐵 𝐴
“AND” INTERSECTIONS (∩)
INDEPENDENT EVENT
Example: Suppose you spin each of these spinners. What is the probability of
spinning a star and a “B”?
3
P(star) =
(3 stars out of 8 outcomes)
8
1
P(B) =
(2 “B”s out of 6 outcomes)
3
3 1
3
1
∙ =
=
P(star, B) =
8 3 24 8
Slide 13
DEPENDENT EVENT
Example: There are 6 black socks and 8 white socks in your dresser drawer. If you
get dressed in the dark and take a sock without looking and then take another sock
without replacing the first, what is the probability that you will get 2 black socks?
P(black first) =
6
3
or
14
7
5
P(black second) =
(There are 13 socks left and 5 are black)
13
Therefore…
P(black, black) =
3 5
15

or
7 13
91
“OR” UNIONS (U)
A
B
MUTUALLY EXCLUSIVE EVENTS
• TWO EVENTS ARE MUTUALLY
EXCLUSIVE IF THEY CANNOT OCCUR
AT THE SAME TIME.
• DISJOINT IS ANOTHER WORD THAT
MEANS MUTUALLY EXCLUSIVE
𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡: 𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 0
• IF TWO EVENTS ARE MUTUALLY
EXCLUSIVE, THEN THE PROBABILITY
OF EITHER OCCURRING IS THE SUM
OF THE PROBABILITIES OF EACH
OCCURRING
𝑃 𝐴 𝑜𝑟 𝐵 = 𝑃 𝐴 + 𝑃(𝐵)
NON-MUTUALLY EXCLUSIVE EVENTS
• IN EVENTS WHICH AREN’T MUTUALLY
EXCLUSIVE, THERE IS SOME
OVERLAP.
• WHEN P(A) AND P(B) ARE ADDED,
THE PROBABILITY OF THE
INTERSECTION (AND) IS ADDED
TWICE.
• TO COMPENSATE FOR THAT DOUBLE
ADDITION, THE INTERSECTION
NEEDS TO BE SUBTRACTED.
𝑃 𝐴 𝑜𝑟 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
COMMON EXAMPLES:
• WHAT ROUTE TO TAKE TO WORK TO
AVOID DELAYS OR ACCIDENTS
• HOW TO DRESS FOR THE WEATHER
(HOT/COLD, RAIN/SNOW, ETC.)
• GAMBLING GAMES OF ALL KINDS
• WHERE TO EAT TO AVOID FOOD
POISONING
CAREER EXAMPLES:
• MEDICAL CAREERS
• ACTUARIAL SCIENCES
• BIO-MATHEMATICIANS
• BIOMEDICAL SCIENCES
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ECOLOGICAL SCIENCES
• FINANCES
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• SO MUCH MORE!!!