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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
Find the probability of obtaining exactly 1 odd number on 4 spins of the arrow 3
2 1
Probability of Two Outcomes
(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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
Find the probability of obtaining exactly 2 odd number on 4 spins of the arrow 3
2 1
Probability of Two Outcomes
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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
Find the probability of obtaining exactly 3 odd number on 4 spins of the arrow 3
2 1
Probability of Two Outcomes
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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
Find the probability of obtaining exactly ? odd number on 4 spins of the arrow 3
2 1
Probability of Two Outcomes
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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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?
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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: Alg. 2 & Trig.
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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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?
( ) ?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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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?
( ) ?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
Aim: Bernoulli Experiment Course: Alg. 2 & Trig.
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: Alg. 2 & Trig.
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.