Binomial

A binomial random variable is the number of successes x in n repeated trials of a binomial experiment.

Binomial Formula: http://stattrek.com/probability-distributions/binomial.aspx

x: The number of successes that result from the binomial experiment.
n: The number of trials in the binomial experiment.
P: The probability of success on an individual trial.
Q: The probability of failure on an individual trial. (This is equal to 1 - P.)
n!: The factorial of n (also known as n factorial).
b(x; n, P): Binomial probability - the probability that an n-trial binomial experiment results in exactly x successes, when the probability of success on an individual trial is P.
_nC_r: The number of combinations of n things, taken r at a time.

Binomial Probability Formula

· b(x; n, P) = _nC_x * P^x * (1 - P)^{n - x}

· b(x; n, P) = {n! / [ x! (n-x)! ] } * P^x * (1 - P)^{n - x}

EXAMPLE #1 is NOT a Binomial because as soon as you pull one camera - the fractions change

Suppose that a box contains 8 cameras and that 3 of them are defective. A sample of 2 cameras is selected at random. Define the random variable X as the number of defective cameras in the sample. Write the probability distribution for X. Give your answers to 2 decimal places.

Because the probabilities do not stay the same from trial to trial (as soon as you remove a phone, the next probability of drawing a defective phone has changed (sampling w/o replacement), this is NOT a binomial distribution.

The probability of 0 defectives is P(0) = 5/8*4/7 = .625 * .571429 = .357143

The probability of 1 defective is P(1) = 3/8*5/7 PLUS " + " 5/8*3/7

Two ways to draw a single defective phone and a single working phone

The probability of 2 defectives is P(2) = 3/8*2/7

The Probability Distribution for X or

The Expected Value E(X) is the sum of x* p(x) = 1*.535714285 + 2*.107142857 = .75

IF this was a Binomial then we would solve it like this: these are the numbers that I get from your example above:

· x=0, 1, 2 Getting 0 working cameras, Getting 1 working camera, Getting 2 working cameras

· k=2 (sometimes this is denoted as n=number of tries/draws/throws/etc.)

· P =.625 Probability of a working camera = P(working camera) = 5/8 = .625

· Q=.375 Probability of a defective camera = Q(defective camera)= 3/8 = .375

The formula is P(k success in n trials) = nCk = ( ⁿ _k) * p ^k * q ^n-k

₂C₀ = ( ² ₀) .625 ⁰ * .375 ^2-0 =1* 1 *.140625 =.140625

₂C₁ = ( ² ₁) .625 ¹ * .375 ^2-1 =2*.62500*.375000 =.468750

₂C₂ = ( ² ₂) .625 ² * .375 ^2-2 =1*.390625 * 1 =.390625

If you add these three results together 0.140625+0.46875+0.390625 they = 1

Binomial Calculator: http://stattrek.com/online-calculator/binomial.aspx

A similar example using 8 cameras with 3 good and 5 defective

P(x = 0) + P(x = 1) + P(x = 2) = must always = 1

P(x = 0) 1st camera pulled has no defects (3 good, 5 defective) 3/8, 2nd camera has no defects, but there are only 7 cameras 2/7

P(x = 0 ) 3/8 * 2/7 = 6 /56 = 0.107143 = 0.11

P(x = 1) 1st camera pulled has no defects (3 good, 5 defective) 3/8, 2nd camera has defects, but there are only 7 cameras 5/7

P(x = 1 ) 3/8 * 5/7 = 15 /56 = 0.267857 = 0.27

I'll let you calculate P(x =2).

Your question about the second part I don't know how to calculate the table or the expected value for X

is a way to do the above calculations neatly on a spreadsheet - so that you don't get things mixed up.

If you use this type of a grid/table in your spreadsheet

I think you will correct your minor issues.

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EXAMPLE #2

A high school baseball player has 0.214 batting average. In one game, he gets 7 at bats. What is the probability he will get at least 2 hits in the game?

P(X greater than or equal to 2) = ans = .46144337117

Binomial Formula:

x=2 : The number of successes that result from the binomial experiment.
n=7 : The number of trials in the binomial experiment.
P=.214 : The probability of success on an individual trial.
Q=(1-.214) : The probability of failure on an individual trial. (This is equal to 1 - P.)
n!=7*6*5*4*3*2*1: The factorial of n (also known as n factorial).
b(x; n, P): Binomial probability - the probability that an n-trial binomial experiment results in exactly x successes, when the probability of success on an individual trial is P.
_nC_x: The number of combinations of n things, taken x at a time.

Binomial Probability Formula b(x; n, P) = _nC_x * P^x * (1 - P)^{n - x}

^·The formula is P(x success in n trials) = nCx = ( ⁿ _x) p ^x * q ^n-x

· b(x; n, P) = {n! / [ x! (n-x)! ] } * P^x * (1 - P)^{n - x}

_nC_x * P^x * (1 - P)^{n - x}

{n! / [ x! (n-x)! ] } * P ^x * (1 -P)^{n - x}

7! / 2!(7-2)!

7.6.~~5.4.3.2.1~~ / 2*1*~~5*4*3*2*1~~ = 7 *6 / 2*1 = 42 / 2 = 21

21 * P ^x * (1 -P)^{n - x}

21 * (.214^2) * ( 1-.214)^7-2)

21 * .045796 * (.786)^5

21 * .045796 * .299994

.288509136960508 This is ONLY for P(x=2)

this question is asking for P(x>=2)

the cumulative Probability is

P(x>=2) = P(x=2) + P(x=3) + P(x=4) + P(x=5) + P(x=6) + p(x=7)

NOTE: to check your answers please verify with the Binomial Calculator http://stattrek.com/online-calculator/binomial.aspx#cumprob

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EXAMPLE #3

A poll shows that 51.6% of American say they believe that the statistic teacher knows the true meaning of life. What is the probability of randomly selecting someone who does not believe that statistic teacher knows the true meaning of life?

Binomial Formula:

x=12 : The number of successes that result from the binomial experiment.
n=1 : The number of trials in the binomial experiment.
P=.516 : The probability of success on an individual trial.
Q=(1-.516) : The probability of failure on an individual trial. (This is equal to 1 - P.)
n!=1: The factorial of n (also known as n factorial).
b(x; n, P): Binomial probability - the probability that an n-trial binomial experiment results in exactly x successes, when the probability of success on an individual trial is P.
_nC_x: The number of combinations of n things, taken x at a time.