Probability Distribution And Approximation SS3 Further Mathematics Lesson Note

• BINOMIAL PROBABILITY DISTRIBUTION
• POISSON PROBABILITY DISTRIBUTION

Probability distribution deals with theoretical probability models based on the randomness of certain natural occurrences.  The binomial and Poisson distribution are discrete.

BINOMIAL DISTRIBUTION

This arises from a repeated random experiment which has two possible outcomes. 

The two possible outcomes of the random experiment are usually called success and failure.

Prob( success)  = P,  Prob(failure) = q

Since the two events are complementary, hence  p+ q = 1  or  p = 1-q, q = 1 – p

The probability of success or failure of an event is the same for each trial and does not influence the probability of success or failure of another trial of the same event.

:. The binomial distribution of n trials and r required outcome(s) is defined as :

       pr(x = r)  = nCrPrqn-r

       whennCr =    n!

                            (n-r)! r!.

The binomial distribution is suitable when the number of trials is not too large.

 Example:

1. Find the probability that when two fair coins are tossed 5 times a head and a tail appear three times.

Solution:

Two fair coins = (HT, TH, TT, HH) = 4

Prob (a head and a tail) = 2/4  = ½

i.e p = ½ , q = ½  (p + q = 1)

n = 5, r = 3.

:. P(x = r)  =nCrprqn-r

p ( x = 3)  = 5C3 ( ½ ) 3  ( ½ ) 5-3

p (x = 3) = 10 x 1/8 x ¼  = 10/32 = 5/16

p (x = 3)  = 0.3125.

 2. It is known that 2 out of every 5 cigarette smokers in a village have cancer of the lungs. Find the probability that out of a random sample of 8 smokers from the village, 5 will have cancer of the lungs.

 Solution

Prob( a smoker has cancer) = 2/5i.e p = 2/5

Prob( a smoker doesn’t have cancer) = 1 – 2/5 = 3/5

:. q = 3/5

  n = 8,   r = 5

Prob(x = -5)  =8C5 (2/5)5  (3/5)3

= 56 x 32 x   27  = 48384

       3125   125    390625

Prob (x = 5)   = 0.124.

EVALUATION

Find the probability that when a fair six-faced die is tossed six times, a prime number appears exactly four times.

POISSON DISTRIBUTION:  The Poisson distribution is more suitable when the number of trials is very large and the probability of success is small. It is defined as:

      Pr(x) =   λx e– λ  , x = 0, 1, 2, 3,

x!

Where   λ   = np                        e = 2.718

P = probability of success, n = number of trials.

 Example:

If 8% of articles in a large consignment are defective, what is the chance that 30 articles selected at random will contain fewer than 3 defective articles?

 Solution

P = 8/100 = 0.08, n = 30

:.  λ   = np = 0.08 x 30 = 2.4.

Prob( fewer than 3)  i.eprob( (0)  +   prob (1)  + prob (2)

Prob (x = 0)  =2.4o  x e-2.4   =  1 x e-2.4

                             0!

Prob(x = 1) = 2.4o  x e-2.4   =  2.4 x e-2.4

                           1!

Prob(x = 2) =2.4o  x e-2.4   =  2.88 x e -2.4

                            2!

Prob(x <3)   =   e -2.4 + 2.4 x e-2.4 + 2.88 x e-2.4

                   = e-2.4 (1 + 2.4 + 2.88)

                   = e-2.4 x   6.28.

 EVALUATION

The probability that a person gets a reaction from a new drug on the market is 0.001. If 200 people are treated with this drug.  Find approximately, the probability that:

1. exactly three persons will get a reaction
2. more than two people will get a reaction

 Properties of Binomial and Poisson Distribution.

 Binomial

It assigns probability to non-occurrence of events i.eProb( x = 0)

Mean  µ  = np

Standard deviation, r = √npq

Variance ð2 = npq

 Poisson

It assigns probability to non-occurrence of events i.eProb(x = 0)

Mean µ = λ = np

Standard deviation, ð = √λ  = √np

Variance, ð2 =   λ   = np

Example:

1. In the probability of tossing a fair coin three times, a head shows up twice. Find the mean and standard deviation.

 Solution

n = 3  Prob(a head)  = ½ , i.e p = ½ ,  q = ½

1. Mean µ = np   = 3 x ½   = 3/2
2. Standard deviation :r = √npq = 3 x ½  x ½  = ¾

 Example

2. 0.2% of the cooks produced by a machine were found to be defective.  If there are 1000 corks, find the mean and standard deviation.

 Solution

P = 0.2%   = 0.002.

N = 1000

1. mean µ = λ = np  = 1000 x 0.002  = 2
2. r = √np  = √2

EVALUATION

In an examination, 60% of the candidates pass.  If 10 candidates were sampled. Find the mean, standard deviation and variance of the candidates.

 GENERAL EVALUATION

1. The probability that a person gets a reaction from a new drug in the market is 0.001.  If 2000 people were treated with this drug, find the mean and standard deviation.
2. 1. In an examination, 60% of the candidates passed.  Use the binomial distribution to calculate the probabilities that a random sample of 10 candidates contains exactly 2 failures.

 READING ASSIGNMENT

• Read probability distribution and further maths. Project 3 from pages 198-201.

 WEEKEND ASSIGNMENT

1. What is the variance of a binomial distribution?      (a) np        (b) √npq           ( c ) npq        (d) p2
2. The mean (µ) of a Poisson distribution is the same as     (a) Standard deviation (b) variance (c) mean   (d) mean deviation
3. If the number of trials is 100 and the probability of success is 0.0001, what is the variance of this distribution (a) 0.00999           (b) 0.1       (c ) 0.01     (d) 0.001
4. If the birth of a male child and that of a female child are equiprobable. Find the probability that in a family of five children, exactly 3 will be male.  (a) 16/5 (b) 5/16 (c) 5/32  (d) 5/21
5. If an unbiased die is thrown repeatedly, what are the chances that the first, six to be thrown will be the third throw? (a)   25/216            (b) 1/6       (c) 25/36     (d)25/31

 THEORY

1. 20% of the total production of transistors produced by a machine are below standard. If a random sample of 6 transistors produced by the machine is taken, what is the probability of getting

(i) exactly 2 (ii) exactly 1   (iii) at least 2  (iv) at most 2   standard transistors?

2. A fair die is thrown five times. Calculate correctly to 3 decimal places, the probability of obtaining

(a) at most two sixes   (b) exactly three sixes