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Aim: Who is Bernoulli and what is his
experiment regarding probability?
Do Now:
Find the probability of obtaining exactly
2 odd numbers on successive spins.
2
1
3
2 2 4
P (2 odd numbers in 2 spins) 

3 3 9
Aim: Bernoulli Experiment
Course: Math Lit.
Probability of Two Outcomes
Find the probability of obtaining exactly
3 odd numbers on successive spins.
2
1
P (3 odd numbers in 3 spins)
2 2 2 8


3 3 3 27
3
4 odd numbers on successive spins.
P (4 odd numbers in 4 spins)
2 2 2 2 16


3 3 3 3 81
Aim: Bernoulli Experiment
Course: Math Lit.
Probability of Two Outcomes
Find the probability of obtaining
exactly 1 odd number on 4 spins
of the arrow
P (odd on 1st spin)
2 1 1 1 2


3 3 3 3 81
P (odd on 3rd spin)
1 1 2 1 2


3 3 3 3 81
2
1
3
P (odd on 2nd spin)
1
3
1 2 1 1 2
2
1
   


 3  3
   
3 3 3 3 81
P (odd on 4th spin)
1 1 1 2 2


3 3 3 3 81
4 possible ways to spin exactly 1 odd number
1
3
8
 2  1
4     
81
 3  3
Aim: Bernoulli Experiment
4
4
4 C1 
1
Course: Math Lit.
Probability of Two Outcomes
Find the probability of obtaining
exactly 2 odd number on 4 spins
of the arrow
P (odd on 1st 2 spins)
O O E
2
2
2 2 1 1  2 1
4
O E O

    
3 3 3 3  3  3
81 O E E
6 possible ways of
E O O
spinning exactly 2 odd
E O E
combination
E E O
4 3
of 2 odd
6
2
2
4 C2 
 2  1
numbers out
21




4 C2  
of 4 spins
 3  3
2
1
3
Aim: Bernoulli Experiment
Course: Math Lit.
E
E
O
E
O
O
24
81
Probability of Two Outcomes
Find the probability of obtaining
exactly 3 odd number on 4 spins
of the arrow
P (odd on 1st 3 spins)
3
2
1
2 2 2 1  2 1
8

    
3 3 3 3  3   3  81
combination
4 3 2
of 3 odd
4
4 C3 
numbers out
3 21
of 4 spins
3
1
24
 2  1




4 C3  
81
 3  3
Aim: Bernoulli Experiment
Course: Math Lit.
1
3
Probability of Two Outcomes
Find the probability of obtaining
exactly ? odd number on 4 spins
of the arrow
2
1
3
What are the two outcomes here?
odd and not odd
4 possible ways to spin
exactly 1 odd number
1
3
2
 2  1
4    
81
 3  3
6 possible ways to spin
exactly 2 odd numbers
2
2
24
 2  1




4 C2  
81
 3  3
4 possible ways to spin
exactly 3 odd numbers
probability experiments
w/exactly 2 possible
outcomes binomialCourse:
probability
or
a
Math Lit.
Bernoulli experiment
3
1
24
 2  1




4 C3  
Experiment
3 Bernoulli81
 3   Aim:
Combinations & Pascal’s Triangle
0C0 =
1C0 =
2C 0 =
3C0 =
4C0 =
1
1
3C1 =
1
C
10
5 2=
1
1
1
1
5
7
21
3C2 =
16
1
3C3 =
4
1
4C4 =
1 10 C
C
5 3=
5 4= 5
2
1
3
10
10
Aim: Bernoulli Experiment
5C5 =
1
5
15
35
1
1
4
20
35
3
4C3 =
6
15
1
2C2 =
2
3
4
6
1C1 =
3
4C2 =
1
1
1
2C1 =
1 4C 1 = 4
5C0 = 1 5C1 = 5
1
1
6
21
7
Course: Math Lit.
1
1
1
Probability of Success & Failure
Consider n independent trials of an
experiment where each trial has two possible
outcomes called success and failure. The
probability of success on each trial is p and
the probability of failure is q = 1 – p .
kqn – k
C
p
n k
gives the probability of k successes in
the n trials of the experiment
A fair coin is tossed seven times.
Find the probability of obtaining
four heads.
Aim: Bernoulli Experiment
Course: Math Lit.
Probability of Success & Failure
Consider n independent trials of an experiment where each trial has
two possible outcomes called success and failure. The probability of
success on each trial is p and the probability of failure is q = 1 – p.
kqn – k
C
p
n k
gives the probability of k successes in the n trials of the experiment
A fair coin is tossed seven times. Find the
probability of obtaining four heads.
What is p? 1/2 What is q? q = 1 – p = 1/2
What is n? 7
4
1 1
  
7 C4 
 2  2
Aim: Bernoulli Experiment
What is k? 4
74
35

128
Course: Math Lit.
Model Problem
If a fair coin is tossed 10 times, what is the
probability that it falls tails exactly 6 times?
kqn – k
C
p
n k
probability of success p = P(tails) = ½
probability of failure q = P(not tails) = ½
n = 10 trials
r = 6 number of successes
6
1 1



10 C 6 
 2  2
10 6
10  9  8  7  6  5  1   1 





6 5 4 3 21  2   2 
1 1
210 105
 210  


64 16 1024 512
Aim: Bernoulli Experiment
6
Course: Math Lit.
4
Model Problem
If 5 fair dice are tossed, what is the
probability that they show exactly 3 fours?
kqn – k
C
p
n k
probability of success p = P(4) = 1/6
probability of failure q = P(not 4) = 5/6
n = 5 trials
r = 3 number of successes
3
1  5
  6
5 C3 
6
   
5 3
54 3 1   5 

3  2  1  6   6 
1 25 250
125
 10 



216 36 7776 3888
Aim: Bernoulli Experiment
3
2
Course: Math Lit.
Aim: Who is Bernoulli and what is his
experiment regarding probability?
Do Now:
If 6 fair dice are tossed, what is the
probability that they show exactly 4 threes?
Aim: Bernoulli Experiment
Course: Math Lit.
At least
David is rolling 5 dice. To win at this game,
at least three of the 5 dice must be “ones”.
David, therefore, can win by rolling 3, 4 or 5
“ones”. What is the probability David can
win?
In General:
At least r successes in n trials means r,
r + 1, r + 2, . . . , n successes.
Probabilities must be added
Aim: Bernoulli Experiment
Course: Math Lit.
At Least
Rose is the last person to compete in a
basketball free-throw contest. To win, Rose
must be successful in at least 4 out of 5 throws.
If the probability that Rose will be successful
on any single throw is 3/4, what is the
probability that Rose will win the contest?
Rose must be successful in 4 or in 5 throws.
3
1
P  success  
P  failure  
4
4
4
1
405
 3 1
P (4 of 5 successes)  5 C4     
 4   4  1024
Aim: Bernoulli Experiment
Course: Math Lit.
At Least
Rose is the last person to compete in a basketball freethrow contest. To win, Rose must be successful in at
least 4 out of 5 throws. If the probability that Rose
will be successful on any single throw is 3/4, what is
the probability that Rose will win the contest?
5
0
+
P(5 of 5)
243
 3 1
P (5 of 5 successes)  5 C5     
 4   4  1024
P(at least 4 out of 5 successes) =
P(4 of 5)
4
1
5
3  1
3 1


P (at least 4)  5 C4      5 C5    
 4  4
 4  4
405
243
+
1024
1024
81
Aim: Bernoulli Experiment
= Course: Math Lit.
128
0
At Most
A family of 5 children is chosen at random.
What is the probability that there are at most
2 boys in this family of 5 children?
Success means to have 0, 1, or 2 boys.
1
1
P  boy  
P  girl  
2
2
0
5
1
1 1
no boys:
P (0)  5 C0     
32
 2  2
1
4
5
1 1
P (1)  5 C1     
32
 2  2
1 boy:
2
2 boys:
3
10 5
1 1
P (2)  5 C 2     

32 16
 2  2
Aim: Bernoulli Experiment
Course: Math Lit.
At Most
A family of 5 children is chosen at random. What is
the probability that there are at most 2 boys in this
family of 5 children?
Success means to have 0, 1, or 2 boys.
P(at most 2 boys out of 5 children) =
P(0 of 5)
0
+
P(1 of 5)
5
1
+
P(2 of 5)
4
2
1 1
1 1
1
 5 C1      5 C 2  



5 C0 
 2  2
 2  2
 2
1
5
10
+
+
32
32
32
1

2
Aim: Bernoulli Experiment
Course: Math Lit.
1
 2
 
3
At least
A coin is loaded so that the probability of
heads is 4 times the probability of tails.
a. What is the probability of heads on a single
throw?
b. What is the probability of at least 1 tail in 5
throws?
4
P ( heads )  ?
5
4x
1
P ( tails )  ?
5
+
x
P(heads) = 4x/5x = 4/5
P(tails) = x/5x = 1/5
Aim: Bernoulli Experiment
Course: Math Lit.
= 5x
At least
A coin is loaded so that the probability of heads is 4
times the probability of tails.
b. What is the probability of at least 1 tail in 5
throws?
4
1
P ( heads )  ?
P ( tails )  ?
5
5
P(at least 1 tail in 5 throws) =
P(1 t) + P(2 t) + P(3 t) + P(4 t) + P(5 t)
1
4
2
3
3
 1  4
1  4
1  4
 5 C1      5 C 2      5 C 3    
 5  5
 5  5
 5  5
4
1
5
0
2
2101
 1  4
1  4
 5 C4      5 C5     
3125
 5  5
 5  5
Aim: Bernoulli Experiment
Course: Math Lit.
Alternate Solution
A coin is loaded so that the probability of heads is 4
times the probability of tails.
b. What is the probability of at least 1 tail in 5
throws? success
or what is Probability of all heads? failure
P(success) = 1 – P(failure)
4
1
P ( heads )  ?
P ( tails )  ?
5
5
5
0
1024
 4  4
P(all heads)  5 C5     
3125
 5  5
P (at least 1 tail )  1  P (all heads )
1024 2101
 1

Aim: Bernoulli Experiment
Math Lit.
3125 Course:
3125
4 pt. Regents Question
The probability that the Stormville Sluggers
will win a baseball game is 2/3. Determine the
probability, to the nearest thousandth, that the
Stormville Sluggers will win at least 6 of their
next 8 games.
Aim: Bernoulli Experiment
Course: Math Lit.
4 pt. Regents Question
A study shows that 35% of the fish caught in a
local lake had high levels of mercury. Suppose
that 10 fish were caught from this lake. Find,
to the nearest tenth of a percent, the probability
that at least 8 of the 10 fish caught did not
contain high levels of mercury.
Aim: Bernoulli Experiment
Course: Math Lit.
The Product Rule
Aim: Bernoulli Experiment
Course: Math Lit.
The Product Rule
Aim: Bernoulli Experiment
Course: Math Lit.
The Product Rule
Aim: Bernoulli Experiment
Course: Math Lit.