Aim: Who is Bernoulli and what is his experiment regarding probability?

Aim: Bernoulli Experiment Course: Alg. 2 & Trig.

Do Now:

Aim: Who is Bernoulli and what is his experiment regarding probability?

Find the probability of obtaining exactly 2 odd numbers on successive spins.

3

2 1

2 2 4(2 odd numbers in 2 spins)

3 3 9P


Find the probability of obtaining exactly 3 odd numbers on successive spins.

3

2 1

Probability of Two Outcomes

(3 odd numbers in 3 spins)

2 2 2 8

3 3 3 27

P

4 odd numbers on successive spins.

(4 odd numbers in 4 spins)

2 2 2 2 16

3 3 3 3 81

P


Find the probability of obtaining exactly 1 odd number on 4 spins of the arrow 3

2 1


(odd on 1st spin)

2 1 1 1 2

3 3 3 3 81

P

(odd on 2nd spin)

1 2 1 1 2

3 3 3 3 81

P

(odd on 3rd spin)

1 1 2 1 2

3 3 3 3 81

P

(odd on 4th spin)

1 1 1 2 2

3 3 3 3 81

P

4 possible ways to spin exactly 1 odd number1 3

2 1 84

3 3 81

1 32 1

3 3

4 1

44

1C



2 1


2 2

(odd on 1st 2 spins)

2 2 1 1 2 1 4

3 3 3 3 3 3 81

P

O O E E

O E O E

O E E O

E O O E

E O E O

E E O O

6 possible ways of spinning exactly 2 odd

combination of 2 odd numbers out of 4 spins

4 2

4 36

2 1C

2 2

4 2

2 1 24

3 3 81C



2 1


3 1

(odd on 1st 3 spins)

2 2 2 1 2 1 8

3 3 3 3 3 3 81

P

combination of 3 odd numbers out of 4 spins

4 3

4 3 24

3 2 1C

3 1

4 3

2 1 24

3 3 81C


Find the probability of obtaining exactly ? odd number on 4 spins of the arrow 3

2 1


What are the two outcomes here?

odd and not odd

4 possible ways to spin exactly 1 odd number

1 32 1 2

43 3 81

2 2

4 2

2 1 24

3 3 81C

6 possible ways to spin exactly 2 odd numbers

4 possible ways to spin exactly 3 odd numbers

3 1

4 3

2 1 24

3 3 81C

probability experiments w/exactly 2 possible

outcomes - binomial probability or a

Bernoulli experiment


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 .

nCk pkqn – k

gives the probability of k successes in the n trials of the experimentA fair coin is tossed seven times. Find the probability of obtaining four heads.

q = 1 – p


Probability of Success & FailureConsider 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.

nCk pkqn – 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 q = 1 – p = 1/2

What is k? 4What is n? 7

What is q?

4 7 4

7 4

1 1

2 2C

35

128


Model Problem

If a fair coin is tossed 10 times, what is the probability that it falls tails exactly 6 times?

probability of success p = P(tails) = ½nCk pkqn – k

probability of failure q = P(not tails) = ½

n = 10 trials

r = 6 number of successes6 10 6 6 4

10 6

1 1 10 9 8 7 6 5 1 1

2 2 6 5 4 3 2 1 2 2

1 1 210 105210

64 16 1024 512

C


Model Problem

If 5 fair dice are tossed, what is the probability that they show exactly 3 fours?

probability of success p = P(4) = 1/6nCk pkqn – k

probability of failure q = P(not 4) = 5/6

n = 5 trials

r = 3 number of successes3 5 3 3 2

5 3

1 5 5 4 3 1 5

6 6 3 2 1 6 6

1 25 250 12510

216 36 7776 3888

C


Do Now:

Aim: Who is Bernoulli and what is his experiment regarding probability?

If 6 fair dice are tossed, what is the probability that they show exactly 4 threes?


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


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

4 4P success P failure

4 1

5 4

3 1 405(4 of 5 successes)

4 4 1024P C


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?

5 0

5 5

3 1 243(5 of 5 successes)

4 4 1024P C

P(at least 4 out of 5 successes) =

P(4 of 5) + P(5 of 5)

(at least 4)P

405 243 +

1024 102481

=128

4 1

5 4

3 1

4 4C

5 0

5 5

3 1

4 4C


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

2 2P boy P girl

no boys:0 5

5 0

1 1 1(0)

2 2 32P C

1 boy:1 4

5 1

1 1 5(1)

2 2 32P C

2 boys:2 3

5 2

1 1 10 5(2)

2 2 32 16P C


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) + P(1 of 5) + P(2 of 5)

1 5 10 + +

32 32 321

2

0 5

5 0

1 1

2 2C

1 4

5 1

1 1

2 2C

2 3

5 2

1 1

2 2C


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?

( ) ?P heads ( ) ?P tails

4x x+ = 5x

P(heads) = 4x/5x = 4/5

P(tails) = x/5x = 1/5

4

5

1

5


At least



( ) ?P heads ( ) ?P tails 4

5

1

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

5 1

1 4

5 5C

2 3

5 2

1 4

5 5C

3 2

5 3

1 4

5 5C

4 1

5 4

1 4

5 5C

5 0

5 5

1 4

5 5C

2101

3125


Alternate Solution



( ) ?P heads ( ) ?P tails 4

5

1

5

P(all heads) 5 0

5 5

4 4 1024

5 5 3125C

or what is Probability of all heads?

P(success) = 1 – P(failure)

( 1 ) 1 ( )

1024 21011

3125 3125

P at least tail P all heads

success

failure


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.


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.

Documents

Aim: Who is Bernoulli and what is his experiment regarding probability?