Probability II

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Probability II
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Probability

• Denoted by P(Event)

This method for calculating probabilities is only
appropriate when the outcomes of the sample space
are equally likely.
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Experimental Probability

• The relative frequency at which a chance
  experiment occurs
• Flip a fair coin 30 times get 17 heads

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Law of Large Numbers

• As the number of repetitions of a chance
  experiment increase, the difference between the
  relative frequency of occurrence for an event and
  the true probability approaches zero.

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Basic Rules of Probability

• Rule 1. Legitimate Values
• For any event E,
• 0 lt P(E) lt 1
• Rule 2. Sample space
• If S is the sample space,
• P(S) 1

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Rule 3. Complement For any event E, P(E)
P(not E) 1
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Independent

• Two events are independent if knowing that one
  will occur (or has occurred) does not change the
  probability that the other occurs
• A randomly selected student is female - What is
  the probability she plays soccer for FHS?
• A randomly selected student is female - What is
  the probability she plays football for FHS?

Independent
Not independent
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Rule 4. Multiplication If two events A B are
independent, General rule
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What does this mean?
Independent?
Yes
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Ex. 1) A certain brand of light bulbs are
defective five percent of the time. You randomly
pick a package of two such bulbs off the shelf of
a store. What is the probability that both bulbs
are defective? Can you assume they are
independent?
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Independent?
Yes
No
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• Ex. 2) There are seven girls and eight boys in a
  math class. The teacher selects two students at
  random to answer questions on the board. What is
  the probability that both students are girls?
• Are these events independent?

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Ex 6) Suppose I will pick two cards from a
standard deck without replacement. What is the
probability that I select two spades?
Are the cards independent?
NO
P(A B) P(A) ? P(BA)
Read probability of B given that A occurs
P(Spade Spade) 1/4 ? 12/51 1/17
The probability of getting a spade given that a
spade has already been drawn.
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Rule 5. Addition If two events E F are
disjoint, P(E or F) P(E) P(F) (General) If
two events E F are not disjoint, P(E or F)
P(E) P(F) P(E F)
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What does this mean?
Mutually exclusive?
Yes
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Ex 3) A large auto center sells cars made by many
different manufacturers. Three of these are
Honda, Nissan, and Toyota. Suppose that P(H)
.25, P(N) .18, P(T) .14. What is the
probability that the next car sold is a Honda,
Nissan, or Toyota?
Are these disjoint events?
yes
P(H or N or T)
.25 .18 .14 .57
P(not (H or N or T)
1 - .57 .43
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Mutually exclusive?
Yes
No
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Ex. 4) Musical styles other than rock and pop are
becoming more popular. A survey of college
students finds that the probability they like
country music is .40. The probability that they
liked jazz is .30 and that they liked both is
.10. What is the probability that they like
country or jazz?
P(C or J) .4 .3 -.1 .6
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Mutually exclusive?
Yes
No
Independent?
Yes
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Ex 5) If P(A) 0.45, P(B) 0.35, and A B are
independent, find P(A or B).
Is A B disjoint?
NO, independent events cannot be disjoint
If A B are disjoint, are they
independent? Disjoint events do not happen at the
same time. So, if A occurs, can B occur?
Disjoint events are dependent!
P(A or B) P(A) P(B) P(A B)
If independent, multiply
How can you find the probability of A B?
P(A or B) .45 .35 - .45(.35) 0.6425
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If a coin is flipped a die rolled at the same
time, what is the probability that you will get a
tail or a number less than 3?
Sample Space
H1 H2 H3 H4 H5 H6 T1
T2 T3 T4 T5 T6
Flipping a coin and rolling a die are independent
events. Independence also implies the events
are NOT disjoint (hence the overlap). Count T1
and T2 only once!
P (heads or even) P(tails) P(less than 3)
P(tails less than 3)
1/2 1/3 1/6 2/3
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• Ex. 7) A certain brand of light bulbs are
  defective five percent of the time. You randomly
  pick a package of two such bulbs off the shelf of
  a store. What is the probability that exactly one
  bulb is defective?

P(exactly one) P(D DC) or P(DC D)
(.05)(.95) (.95)(.05)
.095
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Ex. 8) A certain brand of light bulbs are
defective five percent of the time. You randomly
pick a package of two such bulbs off the shelf of
a store. What is the probability that at least
one bulb is defective?
P(at least one) P(D DC) or P(DC D) or (D
D) (.05)(.95) (.95)(.05)
(.05)(.05) .0975
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Rule 6 Conditional Probability

• A probability that takes into account a given
  condition

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Ex 10) In a recent study it was found that the
probability that a randomly selected student is a
girl is .51 and is a girl and plays sports is
.10. If the student is female, what is the
probability that she plays sports?
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Ex 11) The probability that a randomly selected
student plays sports if they are male is .31.
What is the probability that the student is male
and plays sports if the probability that they are
male is .49?
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Probabilities from two way tables
Stu Staff Total American 107 105 212 Eu
ropean 33 12 45 Asian 55 47 102 Total 195 16
4 359
12) What is the probability that the driver is a
student?
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Probabilities from two way tables
Stu Staff Total American 107 105 212 Eu
ropean 33 12 45 Asian 55 47 102 Total 195 16
4 359
13) What is the probability that the driver
drives a European car?
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Probabilities from two way tables
Stu Staff Total American 107 105 212 Eu
ropean 33 12 45 Asian 55 47 102 Total 195 16
4 359
14) What is the probability that the driver
drives an American or Asian car?
Disjoint?
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Probabilities from two way tables
Stu Staff Total American 107 105 212 Eu
ropean 33 12 45 Asian 55 47 102 Total 195 16
4 359
15) What is the probability that the driver is
staff or drives an Asian car?
Disjoint?
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Probabilities from two way tables
Stu Staff Total American 107 105 212 Eu
ropean 33 12 45 Asian 55 47 102 Total 195 16
4 359
16) What is the probability that the driver is
staff and drives an Asian car?
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Probabilities from two way tables
Stu Staff Total American 107 105 212 Eu
ropean 33 12 45 Asian 55 47 102 Total 195 16
4 359
17) If the driver is a student, what is the
probability that they drive an American car?
Condition
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Probabilities from two way tables
Stu Staff Total American 107 105 212 Eu
ropean 33 12 45 Asian 55 47 102 Total 195 16
4 359
18) What is the probability that the driver is a
student if the driver drives a European car?
Condition
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Example 19 Management has determined that
customers return 12 of the items assembled by
inexperienced employees, whereas only 3 of the
items assembled by experienced employees are
returned. Due to turnover and absenteeism at an
assembly plant, inexperienced employees assemble
20 of the items. Construct a tree diagram or a
chart for this data. What is the probability
that an item is returned? If an item is
returned, what is the probability that an
inexperienced employee assembled it?
P(returned) 4.8/100 0.048
Returned Not returned Total
Experienced 2.4 77.6 80
Inexperienced 2.4 17.6 20
Total 4.8 95.2 100
P(inexperiencedreturned) 2.4/4.8 0.5
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Returned Not returned Total
Experienced 2.4 77.6 80
Inexperienced 2.4 17.6 20
Total 4.8 95.2 100