QuestionsQuestions

Question 5Question 5

There are 2 bags. One Containing 5 There are 2 bags. One Containing 5 white balls and 4 black ball and other white balls and 4 black ball and other containing 4 white and 5 black balls. containing 4 white and 5 black balls. One ball is transferred from one One ball is transferred from one the first bag to the second bag then the first bag to the second bag then a ball is drawn from the second bag. a ball is drawn from the second bag. What is the probability that is a white What is the probability that is a white ball.ball.

41/9041/90

QuestionsQuestions

Random variableRandom variable

QuestionsQuestions

Find the expected value of the heads Find the expected value of the heads when two coins are tossedwhen two coins are tossed

Question no 2Question no 2

A box contains six tickets. Two of the A box contains six tickets. Two of the tickets carry a price of Rs. 5 each tickets carry a price of Rs. 5 each and the other four prices of Re.1and the other four prices of Re.1

A) If one ticket is drawn what is the A) If one ticket is drawn what is the expected value of the price?expected value of the price?

B)If two tickets are drawn ,what is B)If two tickets are drawn ,what is the expected value of the price.the expected value of the price.

QuestionsQuestions

Binomial DistributionBinomial Distribution

The The binomial distributionbinomial distribution describes the describes the behavior of a count variable X if the behavior of a count variable X if the

following conditions apply:following conditions apply: 1:1: The number of observations n is fixed. The number of observations n is fixed. 2:2: Each observation is independent. Each observation is independent. 3:3: Each observation represents one of Each observation represents one of

two outcomes ("success" or "failure"). two outcomes ("success" or "failure"). 4:4: The probability of "success" p is the The probability of "success" p is the

same for each outcome. same for each outcome.

If these conditions are met, then X If these conditions are met, then X has a binomial distribution with has a binomial distribution with parameters n and p, abbreviated parameters n and p, abbreviated B(n,p). B(n,p).

A random variable X is said to follow A random variable X is said to follow Binomial distribution with parameters Binomial distribution with parameters n and p if its probability function is n and p if its probability function is

f(x)= nCf(x)= nCxxppxxqqn-xn-x

Where x= 0,1,2,….nWhere x= 0,1,2,….n

P + q = 1P + q = 1

Mean of binomial distributionMean of binomial distribution

Mean of binomial distribution = npMean of binomial distribution = np n- number of trialsn- number of trials p- probability of successp- probability of success

Standard deviation of a binomial Standard deviation of a binomial distributiondistribution

Standard deviation of a binomial Standard deviation of a binomial distribution = npqdistribution = npq

n- number of trialsn- number of trials p– probability of successp– probability of success q- Probability Of Failure or (1-p)q- Probability Of Failure or (1-p)

Question no 1Question no 1

Four coins are tossed simultaneously. Four coins are tossed simultaneously. What is the probability of getting two What is the probability of getting two headsheads

Question No 2Question No 2

Eight unbiased coins are tossed Eight unbiased coins are tossed simultaneously. Find the probability simultaneously. Find the probability of getting of getting

Exactly four headsExactly four heads No heads at allNo heads at all 6 or more heads6 or more heads Utmost two headsUtmost two heads Number of heads ranging from 3 to 5Number of heads ranging from 3 to 5

Question no 3Question no 3

Eight coins are tossed simultaneously Eight coins are tossed simultaneously 256 times . Find the expected 256 times . Find the expected frequenciesfrequencies

Find mean and Standard DeviationFind mean and Standard Deviation

Home workHome work

The following Data show the number The following Data show the number of seeds germinating out of 10 on of seeds germinating out of 10 on damp filter for 80 sets of seeds. Fit a damp filter for 80 sets of seeds. Fit a binomial distribution of this data and binomial distribution of this data and find the expected frequenciesfind the expected frequencies

Poison DistributionPoison Distribution

Called law of improbable Called law of improbable events-describe the behaviour events-describe the behaviour

of rare eventsof rare events

Discrete Probability distribution

FormulaFormula

The Probability of N success out of n The Probability of N success out of n trials is given by trials is given by

Where x is a discrete random variable assuming values Where x is a discrete random variable assuming values 0,1,2… 0,1,2…

m is called parameter of Poisson distributionm is called parameter of Poisson distribution

ExampleExample

., Find the probability of 4 customers arriving in 3 minutes when the mean is 3.6.

3.6 43.6

.19124!

eP X

Question no 1Question no 1

If If 3%3% of electric bulbs manufactured of electric bulbs manufactured by a company are defective. Find the by a company are defective. Find the probability that in a sample of probability that in a sample of 100100 bulbs exactly bulbs exactly fivefive bulbs are defective bulbs are defective

Question No 2Question No 2

Fit a poison distribution to the Fit a poison distribution to the following data and calculate the following data and calculate the theoretical frequenciestheoretical frequencies

XX 00 11 22 33 44 YY 123123 5959 1414 33 11

Home WorkHome Work

Between the hours 2 and 4 pm the Between the hours 2 and 4 pm the average number of phone calls per average number of phone calls per minute coming into the switch minute coming into the switch board of a company is 2.5. find the board of a company is 2.5. find the probability that during one probability that during one particular minute there will beparticular minute there will be

1)1) No phone call at allNo phone call at all2)2) Exactly two callsExactly two calls3)3) At least Five callsAt least Five calls

Home work Home work

Find the probability that almost 5 Find the probability that almost 5 defective fuses will be found in a box defective fuses will be found in a box of 200 fuses. An experience shows of 200 fuses. An experience shows that 2% of such fuses are defective.that 2% of such fuses are defective.

Normal distributionNormal distribution

Both binomial and Poisson distributions Both binomial and Poisson distributions consist of all the values (finite ) of a consist of all the values (finite ) of a random variable that made up of these random variable that made up of these discrete and associated probabilitiesdiscrete and associated probabilities

Normal probability distribution is one of Normal probability distribution is one of the most frequently used distribution. IT is the most frequently used distribution. IT is normally described in terms of continuous normally described in terms of continuous curve in the shape of a bell (symmetrical)curve in the shape of a bell (symmetrical)

ExampleExample

The weekly wages of 1,000 workmen The weekly wages of 1,000 workmen are normally distributed around a are normally distributed around a mean of Rs. 70 and with a standard mean of Rs. 70 and with a standard deviation of Rs.5 Estimate the number deviation of Rs.5 Estimate the number of workers whose weekly wages will of workers whose weekly wages will bebe

Between Rs.70 and Rs.72Between Rs.70 and Rs.72 Between Rs.69 and Rs.72Between Rs.69 and Rs.72 More than Rs.75More than Rs.75 Less than Rs. 63Less than Rs. 63

Home work 1Home work 1Normal distributionNormal distribution

Consider a project that yields an average Consider a project that yields an average cash flow of Rs. 500 lakhs with a standard cash flow of Rs. 500 lakhs with a standard deviation of Rs. 60 lakhs. Calculate the deviation of Rs. 60 lakhs. Calculate the following probabilitiesfollowing probabilities Cash flow will be more than Rs. 560 lakhsCash flow will be more than Rs. 560 lakhs Cash flow will be less than Rs.420 lakhsCash flow will be less than Rs.420 lakhs Cash flow will lie between Rs. 460 lakhs and Cash flow will lie between Rs. 460 lakhs and

Rs.540 lakhsRs.540 lakhs Cash flow will be more than Rs. 680 lakhsCash flow will be more than Rs. 680 lakhs

AnswersAnswers

1.Z=(x-1.Z=(x-)/)/=(560-500)/60=1.0=(560-500)/60=1.0

2.z=-1.332.z=-1.33

3.-.66 and .66(2*.2454=.49083.-.66 and .66(2*.2454=.4908

4. 3.0, 0.00134. 3.0, 0.0013