Transcript Probability PowerPoint

VOCAB REVIEW:
Pascal’s triangle - A triangular
arrangement of where each row
corresponds to a value of n.
Pascal’s Triangle
n = 0; 20
This is a Pascal’s Triangle
– Each row is labeled as n
• The first row is n = 0
• Second is n = 1 etc…
– Each term is nCr
• n is the row
• r is the position in the row starting with 0
– Each term is the sum of the two directly above it.
n = 1; 21
n = 2; 22
n = 3; 23
n = 4; 24
1.
The 6 members of a Model UN student club must choose 2
representatives to attend a state convention. Use Pascal’s
triangle to find the number of combinations of 2 members that
can be chosen as representatives.
2.
Use Pascal’s triangle again
to find the number of combinations
of 2 members that can be chosen
if the Model UN club has 7
members.
Binomial Theorem
The Binomial Theorem
• Gives us the coefficients for a binomial expansion
• The values in a row of Pascal's triangle are the
coefficients in a binomial expansion of the same degree
as the row.
• A binomial expansion of degree n is (a + b)n.
• The variables are
anb0 + an-1b1 + … + a1bn-1 + anb0 + a0bn
Expand a Power of a Binomial Sum
x
1st
2
 y 
term
2nd term
Pascal’s #
3
 x   x  xx  1x 
26 3
24 2
1
22
2 0
 y1  y   yy  yy
0
1
1
3
22
33
3
1
x  3x y  3x y  y
6
4
2
2
3
 a  2b 
1st term
2nd term
Pascal’s #
4

 a   aa  a 
aa 
1a 
0
1
2 2
3 3
44
2b 
 21b  -22bb   42bb  -82bb  16
44
1
33
4
22
6
1
0
4
1
a  8a b  24a b  32ab  16b
4
3
2 2
3
4
Use the binomial theorem to write the binomial expansion.
1.
 x  3
2.
2 p  q
5
4
3.
 a  2b 
4
4.
5  2 y 
3
Find a Coefficient in an Expansion
• Find the coefficient of x in the expansion of
p
q n
ax  by
m


where rm/p
r
C
(1
st
)
(2
nd
)
n r
nr
• Find the coefficient of x⁴ in the
expansion of (3x + 2)¹º.
n=
r=
x5
Binomial Formula
1.
Use the binomial formula to find the coefficient of the
10
in the expansion of
 q  3z 
q9 z
2.
Find the coefficient of the x5 in the expansion of (x – 3)7?
3.
Find the coefficient of the x3 in the expansion of (2x +5)8?
term
12.3 An Introduction to Probability
What do you know about probability?
• Probability is a number from 0 to 1 that
tells you how likely something is to
happen.
• Probability can have two main approaches
-experimental probability
-theoretical probability
Experimental vs.Theoretical
Experimental probability:
P(event) = number of times event occurs
total number of trials
Theoretical probability:
P(E) = number of favorable outcomes
total number of possible outcomes
How can you tell which is experimental and which
is theoretical probability?
Experimental:
You tossed a coin 10
times and recorded a
head 3 times, a tail 7
times
P(head)= 3/10
P(tail) = 7/10
Theoretical:
Toss a coin and getting
a head or a tail is 1/2.
P(head) = 1/2
P(tail) = 1/2
Experimental probability
Experimental probability is found by
repeating an experiment and observing
the outcomes.
P(head)= 3/10
A head shows up 3 times out of 10 trials,
P(tail) = 7/10
A tail shows up 7 times out of 10 trials
Theoretical probability
HEADS
TAILS
P(head) = 1/2
P(tail) = 1/2
Since there are only
two outcomes,
you have 50/50
chance to get a
head or a tail.
How come I never get a theoretical value in
both experiments? Tom asked.
• If you repeat the
experiment many
times, the results
will getting closer to
the theoretical
value.
• Law of the Large
Numbers
Experimental VS. Theoretical
54
53.4
53
52
51
50
49
50
49.87
48.4
48
47
46
45
1
48.9
Thoeretical
5-trial
10-trial
20-trial
30-trial
Law of the Large Numbers 101
• The Law of Large Numbers was first
published in 1713 by Jocob Bernoulli.
• It is a fundamental concept for probability and
statistic.
• This Law states that as the number of trials
increase, the experimental probability will get
closer and closer to the theoretical
probability.
http://en.wikipedia.org/wiki/Law_of_large_numbers
Contrast experimental and theoretical
probability
Experimental
probability is the
result of an
experiment.
Theoretical
probability is what
is expected to
happen.
You must show the probability set up, the unreduced fraction,
and the reduced fraction in order to receive full credit.
Geometric Probability
• Geometric probabilities are found by
calculating a ratio of two side lengths,
areas, or volumes according to the
problem.
Find a Geometric
Probability
• You throw a dart at the square board. Your
dart is equally likely to hit any point inside
the board. Are you more likely to get 10
points or 0? (use area)
2 5 10
0
• HW 37: pg 719, 13-43 odd