27
Binomial Experiment A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties: The experiment consists of n repeated trials. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. The probability of success, denoted by P, is the same on every trial. The trials are independent ; that is, the outcome on one trial does not affect the outcome on other trials. Consider the following statistical experiment. You flip a coin 2 times and count the number of times the coin lands on heads. This is a binomial experiment because: The experiment consists of repeated trials. We flip a coin 2 times. Each trial can result in just two possible outcomes - heads or tails. The probability of success is constant - 0.5 on every trial. The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials. Notation The following notation is helpful, when we talk about binomial probability.

Binomial Experiment Revised

Embed Size (px)

Citation preview

Page 1: Binomial Experiment Revised

Binomial Experiment

A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties:

The experiment consists of n repeated trials. Each trial can result in just two possible outcomes. We call one of these outcomes

a success and the other, a failure. The probability of success, denoted by P, is the same on every trial. The trials are independent; that is, the outcome on one trial does not affect the

outcome on other trials.

Consider the following statistical experiment. You flip a coin 2 times and count the number of times the coin lands on heads. This is a binomial experiment because:

The experiment consists of repeated trials. We flip a coin 2 times. Each trial can result in just two possible outcomes - heads or tails. The probability of success is constant - 0.5 on every trial. The trials are independent; that is, getting heads on one trial does not affect

whether we get heads on other trials.

Notation

The following notation is helpful, when we talk about binomial probability.

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.) 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.

nCr: The number of combinations of n things, taken r at a time.

Page 2: Binomial Experiment Revised

Binomial Distribution

A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution (also known as a Bernoulli distribution).

Suppose we flip a coin two times and count the number of heads (successes). The binomial random variable is the number of heads, which can take on values of 0, 1, or 2.

The binomial distribution is presented below.

Number of heads Probability

0 0.25

1 0.50

2 0.25

The binomial distribution has the following properties:

The mean of the distribution (μx) is equal to n * P . The variance (σ2

x) is n * P * ( 1 - P ). The standard deviation (σx) is sqrt[ n * P * ( 1 - P ) ].

Page 3: Binomial Experiment Revised

When do we get a Binomial distribution?

The following are the conditions in which probabilities are given by binomial distribution.

1. A trial is repeated 'n' times where n is finite and all 'n' trials are identical. 2. Each trial (or you can call it an event) results in only two mutually exclusive,

exhaustive but not necessarily equally likely possibilities, success or failure. 3. The probability of a "success" outcome is equal to some percentage which is

identified as proportion, p (or p) 4. This proportion p (or p), remains constant throughout all events (or trials). It is

defined as the ratio of the number of successes to the number of trials. 5. The events (or trials) are independent. 6. If probability of success is p or p, then the probability of failures is 1 - p or 1 - p

this is denoted by q. Thus p + q = 1.

Binomial Probability

The binomial probability refers to the probability that a binomial experiment results in exactly x successes. For example, in the above table, we see that the binomial probability of getting exactly one head in two coin flips is 0.50.

Given x, n, and P, we can compute the binomial probability based on the following formula:

Binomial Formula. Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success on an individual trial is P, then the binomial probability is:

b(x; n, P) = nCx * Px * (1 - P)n - x

The Binomial Distribution is one of the discrete probability distribution. It is used when there are exactly two mutually exclusive outcomes of a trial. These outcomes are appropriately labeled Success and Failure. The Binomial Distribution is used to obtain the probability of observing r successes in n trials, with the probability of success on a single trial denoted by p.

Page 4: Binomial Experiment Revised

Example: Suppose a die is tossed 5 times. What is the probability of getting exactly 2 fours?

Solution: This is a binomial experiment in which the number of trials is equal to 5, the number of successes is equal to 2, and the probability of success on a single trial is 1/6 or about 0.167. Therefore, the binomial probability is:

b(2; 5, 0.167) = 5C2 * (0.167)2 * (0.833)3 b(2; 5, 0.167) = 0.161

Cumulative Binomial Probability

A cumulative binomial probability refers to the probability that the binomial random variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper limit).

Example: What is the probability of obtaining 45 or fewer heads in 100 tosses of a coin?

Solution: To solve this problem, we compute 46 individual probabilities, using the binomial formula. The sum of all these probabilities is the answer we seek. Thus,

b(x < 45; 100, 0.5) = b(x = 0; 100, 0.5) + b(x = 1; 100, 0.5) + . . . + b(x = 45; 100, 0.5) b(x < 45; 100, 0.5) = 0.184

Page 5: Binomial Experiment Revised

The binomial formula can also be represented as the following:

P(X = r) = nCr p r (1-p)n-r

where,             n = Number of events.

             r = Number of successful events.             p = Probability of success on a single trial.

             nCr = ( n! / (n-r)! ) / r!             1-p = Probability of failure.

Example: Toss a coin for 12 times. What is the probability of getting exactly 7 heads.

 Step 1: Here,                 Number of trials n = 12                Number of success r = 7 (since we define getting a head as success)                Probability of success on any single trial p = 0.5

  Step 2: To calculate nCr formula is used.            nCr = ( n! / (n-r)! ) / r!                = ( 12! / (12-7)! ) / 7!                = ( 12! / 5! ) / 7!                = ( 479001600 / 120 ) / 5040                = ( 3991680 / 5040 )                = 792

  Step 3: Find pr.            pr = 0.57

               = 0.0078125

  Step 4: To Find (1-p)n-r Calculate 1-p and n-r.            1-p = 1-0.5 = 0.5            n-r = 12-7 = 5

  Step 5: Find (1-p)n-r.            = 0.55 = 0.03125

  Step 6: Solve P(X = r) = nCr p r (1-p)n-r

            = 792 × 0.0078125 × 0.03125            = 0.193359375

The probability of getting exactly 7 heads is 0.19

Page 6: Binomial Experiment Revised

The probability that a random variable X with binomial distribution B(n,p) is equal to the value k, where k = 0, 1,....,n , is given by

, where

. The latter expression is known as the binomial coefficient, stated as "n choose k," or the number of possible ways to choose k "successes" from n observations. For example, the number of ways to achieve 2 heads in a set of four tosses is "4 choose 2", or 4!/2!2! = (4*3)/(2*1) = 6. The possibilities are {HHTT, HTHT, HTTH, TTHH, THHT, THTH}, where "H" represents a head and "T" represents a tail. The binomial coefficient multiplies the probability of one of these possibilities (which is (1/2)²(1/2)² = 1/16 for a fair coin) by the number of ways the outcome may be achieved, for a total probability of 6/16.

Obtaining Coefficients of the Binomial

For obtaining coefficients from the binomial expansion, the following rules may be remembered. To find the terms of the expansion of (q + p)n

1. The first term is qn .2. The second term is nC1q n-1 p.

3. In each succeeding term the power of q is reduced by 1 and the power of p is increased by 1.

4. The coefficient of any term is found by multiplying the coefficient of the preceding term by the power of q in that preceding term, and dividing the products so obtained by one more than the power of p in that preceding term.

Page 7: Binomial Experiment Revised

Pascal's triangle :

0: 11: 1 12: 1 2 13: 1 3 3 14: 1 4 6 4 15: 1 5 10 10 5 16: 1 6 15 20 15 6 17: 1  7  21 35 35 21 7  1 8: 1  8  28 56 70 56 28 8  1 

Row number n contains the numbers for k = 0,…,n. It is constructed by starting with ones at the outside and then always adding two adjacent numbers and writing the sum directly underneath. This method allows the quick calculation of binomial coefficients without the need for fractions or multiplications. For instance, by looking at row number 5 of the triangle, one can quickly read off that

(x + y)5 = 1 x5 + 5 x4y + 10 x3y2 + 10 x2y3 + 5 x y4 + 1 y5.

The binomial distribution for a random variable X with parameters n and p represents the sum of n independent variables Z which may assume the values 0 or 1. If the probability that each Z variable assumes the value 1 is equal to p, then the mean of each variable is equal to 1*p + 0*(1-p) = p, and the variance is equal to p(1-p). By the addition properties for independent random variables, the mean and variance of the binomial distribution are equal to the sum of the means and variances of the n independent Z variables, so

These definitions are intuitively logical. Imagine, for example, 8 flips of a coin. If the coin is fair, then p = 0.5. One would expect the mean number of heads to be half the flips, or np = 8*0.5 = 4. The variance is equal to np(1-p) = 8*0.5*0.5 = 2.

Page 8: Binomial Experiment Revised

Properties of the Binomial distribution :

We get below some important properties of the Binomial distribution without derivations.

1. If x denotes the Binomial variate, expression of x i.e. the mean of the distribution

is given by, 2. The standard deviation of the Binomial distribution is determined by,

3. If in experiment, each of n trials, is repeated N times then expression of r successes i.e. the expected frequency of r successes in N experiment is given by,

Example What is the expression of heads if an unbiased coin is tossed 12 times.?

Solution: Since the expression of x in a binomial distribution is given by, E (x) = np where n = 12 and p = 0.5 . We could expect 12 0.5 = 6 heads.

Example For a Binomial distribution, mean is 2 and standard deviation is 1. Find all the constants of the distribution.

Solution: We are given, Mean () = n p = 2 and S.D. =

Hence the constants of the distribution are n = 4, p = 0.5 and q = 0.5

Page 9: Binomial Experiment Revised

Example: If the probability of a defective bulb is 0.4. Find the mean and the standard deviation for the distribution of the defective bulbs in a lot of 1000 bulbs. What is the expected number of defective bulbs in the lot ?

Solution :We have p = 0.4, n = 1000 and q = 1 - p = 1 - 0.4 = 0.6Mean () = np = 1000 0.2 = 200

Expected number = n p = 0.4 1000 = 400

Example: Take 100 sets of tosses of 10 flips of a fair coin.

In how many cases do you expect to get 7 heads at least?

Solution : We have N = 100 sets. n = 10 trials in each set p = 0.5 and q = 1 - p =0.5 Probability (getting at least 7 heads) in one set

= p (x = 7) + p (x = 8) + p (x = 9) + p (x = 10)

Therefore in 100 sets = N p (r) = 100 (0.171) 17 times you can expect to get at least 7 heads.

Page 10: Binomial Experiment Revised

Example :Six dice are thrown 729 times. How many times do you expect at least three dice to show 5 or 6?

Solution : Let P = probability (showing 5 or 6) = 2/6 = 1/3

q = 1 - p = 1- 1/3 = 2/3

n = 6 and r = 3

Also p (x = r) = probability (at least 3 dice will show 5 or 6 in one trial)

Using the 'complement' theorem

p (x = r) = 1 - [p (x = 0) + p (x = 1) + p (x = 2)]

Example: If the probability of success is How many trials are required in order that

the probability of getting at least one success, is just greater than

Page 11: Binomial Experiment Revised

Solution :

Let 'n' be the required number of trials to get the probability of at least one success which is,

1 - n C 0 P 0 Q n-0 [ since probability (at least one success) = 1 - p (x = 0)

i.e. 1 - probability (No success)]

Example: A and B play a game in which A’s chance of winning is 2/9. In a series of 8 games, what is the chance that A will win at least 9 games ?

Page 12: Binomial Experiment Revised

Solution : Here A’s chance of winning = p = 2/9

Therefore q = 1 - p = 1 - 2/9 = 7/9, n = 8

The probability (A will win at least 6 games in a series of 8 games)

Example: Assume that the probability of a bomb dropped from an aeroplane, striking a target is 1/5. If 6 bombs are dropped, find the probability that

Page 13: Binomial Experiment Revised

(1) exactly two will strike.(2) at least two strikes the target.

Solution :

Page 14: Binomial Experiment Revised

Example: The probability of a man hitting a target is 1/3. How many must he fire so that the probability of hitting the target, at least once, is more than 90% ?

Solution : Here p = 1/3 and q = 2/3 and n = ?

Now, p (hitting the target at least once) > 90%

p ( x 1) = 1 - p (x = 0) must be greater than 90%

Therefore, he must fire at least 6 times so that the probability of hitting the target at least once is more than 90%

Page 15: Binomial Experiment Revised

Example: The probability that a man aged 60 will live to be 70 is 0.65. What is the probability that out of 10 men, now 60, at least 7 will live to be 70?

Solution : P = The probability that a man aged 60 will live to be 70 = 65

q = 1 - P = 1 - 0.65 = 0.35

Number of men ( n ) = 10

Probability that at least 7 men will live to 70

= P ( 7 or 8 or 9 or 10 )

Importance of the Binomial Distribution

The binomial probability distribution is a discrete probability distribution that is useful in describing an enormous variety of real life events. For example, a quality control inspector wants to know the probability of defective light bulbs in a random sample of 10 bulbs if 10 percent of the bulbs are defective. He can quickly obtain the answer from the tables of the binomial probability distribution. The binomial distribution can be used when:

Page 16: Binomial Experiment Revised

1. The outcome or results of each trial in the process are characterized as one of two types of possible outcomes. In other words, they are attributes.

2. The possibility of outcome of any trial does not change and is independent of the results of previous trials.

Negative Binomial Experiment

A negative binomial experiment is a statistical experiment that has the following properties:

The experiment consists of x repeated trials. Each trial can result in just two possible outcomes. We call one of these outcomes

a success and the other, a failure. The probability of success, denoted by P, is the same on every trial. The trials are independent; that is, the outcome on one trial does not affect the

outcome on other trials. The experiment continues until r successes are observed, where r is specified in

advance.

Consider the following statistical experiment. You flip a coin repeatedly and count the number of times the coin lands on heads. You continue flipping the coin until it has landed 5 times on heads. This is a negative binomial experiment because:

The experiment consists of repeated trials. We flip a coin repeatedly until it has landed 5 times on heads.

Each trial can result in just two possible outcomes - heads or tails. The probability of success is constant - 0.5 on every trial. The trials are independent; that is, getting heads on one trial does not affect

whether we get heads on other trials. The experiment continues until a fixed number of successes have occurred; in this

case, 5 heads.

Notation

The following notation is helpful, when we talk about negative binomial probability.

x: The number of trials required to produce r successes in a negative binomial experiment.

r: The number of successes in the negative 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.)

Page 17: Binomial Experiment Revised

b*(x; r, P): Negative binomial probability - the probability that an x-trial negative binomial experiment results in the rth success on the xth trial, when the probability of success on an individual trial is P.

nCr: The number of combinations of n things, taken r at a time.

Negative Binomial Distribution

A negative binomial random variable is the number X of repeated trials to produce r successes in a negative binomial experiment. The probability distribution of a negative binomial random variable is called a negative binomial distribution. The negative binomial distribution is also known as the Pascal distribution.

Suppose we flip a coin repeatedly and count the number of heads (successes). If we continue flipping the coin until it has landed 2 times on heads, we are conducting a negative binomial experiment. The negative binomial random variable is the number of coin flips required to achieve 2 heads. In this example, the number of coin flips is a random variable that can take on any integer value between 2 and plus infinity. The negative binomial probability distribution for this example is presented below.

Number of coin flips Probability

2 0.25

3 0.25

4 0.1875

5 0.125

6 0.078125

7 or more 0.109375

Page 18: Binomial Experiment Revised

Negative Binomial Probability

The negative binomial probability refers to the probability that a negative binomial experiment results in r - 1 successes after trial x - 1 and r successes after trial x. For example, in the above table, we see that the negative binomial probability of getting the second head on the sixth flip of the coin is 0.078125.

Given x, r, and P, we can compute the negative binomial probability based on the following formula:

Negative Binomial Formula. Suppose a negative binomial experiment consists of x

trials and results in r successes. If the probability of success on an individual trial is P,

then the negative binomial probability is:

b*(x; r, P) = x-1Cr-1 * Pr * (1 - P)x - r

Example:

Bob is a high school basketball player. He is a 70% free throw shooter. That means his probability of making a free throw is 0.70. During the season, what is the probability that Bob makes his third free throw on his fifth shot?

Solution: This is an example of a negative binomial experiment. The probability of success (P) is 0.70, the number of trials (x) is 5, and the number of successes (r) is 3.

To solve this problem, we enter these values into the negative binomial formula.

b*(x; r, P) = x-1Cr-1 * Pr * Qx - r b*(5; 3, 0.7) = 4C2 * 0.73 * 0.32

b*(5; 3, 0.7) = 6 * 0.343 * 0.09 = 0.18522

Thus, the probability that Bob will make his third successful free throw on his fifth shot is 0.18522.

Page 19: Binomial Experiment Revised

The formula for negative binomial distribution can also be represented as the following:

P(X = r) = n-1Cr-1 p r (1-p)n-r

where,             n = Number of events.             r = Number of successful events.             p = Probability of success on a single trial.             n-1Cr-1 = ( (n-1)! / ((n-1)-(r-1))! ) / (r-1)!             1-p = Probability of failure.

Example: Find the probability that a man flipping a coin gets the fourth head on the ninth flip.

Step 1: Here,          Number of trials n = 9 (because we flip the coin nine times).          Number of successes r = 4 (since we define Heads as a success).          Probability of success for any coin flip p = 0.5

  Step 2: Find n-1 and r-1.         n-1 = 9-1 = 8         r-1 = 4-1 = 3

  Step 3: To find n-1Cr-1 Calculate ((n-1)-(r-1))!         (n-1)-(r-1) = 8-3 = 5         ((n-1)-(r-1))! = 5! = 120

  Step 4: Find (n-1)!          = 8! = 40320

  Step 5: Find (r-1)!          = 3! = 6

  Step 6: Find (n-1)! / ((n-1)-(r-1))!

Page 20: Binomial Experiment Revised

          = 40320/120 = 336

  Step 7: To Solve n-1Cr-1 formula is used.          = 336/6 = 56

  Step 8: Find pr.          = 0.54 = 0.0625

  Step 9: To Find (1-p)n-r Calculate 1-p and n-r.         1-p = 1-0.5 = 0.5         n-r = 9-4 = 5

  Step 10: Calculate (1-p)n-r.          = 0.55 = 0.03125

  Step 11: Calculate Negative Binomial Distribution.          = 56×0.0625×0.03125 = 0.109375

The probability that the coin will land on heads for the fourth time on the ninth coin flip is 0.1094.

Page 21: Binomial Experiment Revised

RESEARCH AND METHODOLOGY

ASSIGNMENT

ON

BINOMIAL DISTRIBUTION

SUBMITTED TO: SUBMITTED BY:

Ms.APRA ANCHAL AGGARWALLect. in Economics M.Com-IDept. Roll No. 5518

Page 22: Binomial Experiment Revised

APEEJAY COLLEGE OF FINE ARTS