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All Probability

All Probability






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    All Probability All Probability Presentation Transcript

    • Questions
    • Question 5
      • There are 2 bags. One Containing 5 white balls and 4 black ball and other containing 4 white and 5 black balls. One ball is transferred from one the first bag to the second bag then a ball is drawn from the second bag. What is the probability that is a white ball.
      • 41/90
    • Questions Random variable
    • Questions
      • Find the expected value of the heads when two coins are tossed
    • Question no 2
      • A box contains six tickets. Two of the tickets carry a price of Rs. 5 each and the other four prices of Re.1
      • A) If one ticket is drawn what is the expected value of the price?
      • B)If two tickets are drawn ,what is the expected value of the price.
    • Questions Binomial Distribution
    • The binomial distribution describes the behavior of a count variable X if the following conditions apply:
          • 1: The number of observations n is fixed.
          • 2: Each observation is independent.
          • 3: Each observation represents one of two outcomes ("success" or "failure").
          • 4: The probability of "success" p is the same for each outcome.
        • If these conditions are met, then X has a binomial distribution with parameters n and p, abbreviated B(n,p).
      • A random variable X is said to follow Binomial distribution with parameters n and p if its probability function is
      • f(x)= nC x p x q n-x
      • Where x= 0,1,2,….n
      • P + q = 1
    • Mean of binomial distribution
      • Mean of binomial distribution = np
      • n- number of trials
      • p- probability of success
    • Standard deviation of a binomial distribution
      • Standard deviation of a binomial distribution = npq
      • n- number of trials
      • p– probability of success
      • q- Probability Of Failure or (1-p)
    • Question no 1
      • Four coins are tossed simultaneously. What is the probability of getting two heads
    • Question No 2
      • Eight unbiased coins are tossed simultaneously. Find the probability of getting
      • Exactly four heads
      • No heads at all
      • 6 or more heads
      • Utmost two heads
      • Number of heads ranging from 3 to 5
    • Question no 3
      • Eight coins are tossed simultaneously 256 times . Find the expected frequencies
      • Find mean and Standard Deviation
    • Home work
      • The following Data show the number of seeds germinating out of 10 on damp filter for 80 sets of seeds. Fit a binomial distribution of this data and find the expected frequencies
    • Poison Distribution Called law of improbable events-describe the behaviour of rare events Discrete Probability distribution
    • Formula
      • The Probability of N success out of n trials is given by
      • Where x is a discrete random variable assuming values 0,1,2…
      • m is called parameter of Poisson distribution
    • Example
      • ., Find the probability of 4 customers arriving in 3 minutes when the mean is 3.6.
    • Question no 1
      • If 3% of electric bulbs manufactured by a company are defective. Find the probability that in a sample of 100 bulbs exactly five bulbs are defective
    • Question No 2
      • Fit a poison distribution to the following data and calculate the theoretical frequencies
      • X 0 1 2 3 4
      • Y 123 59 14 3 1
    • Home Work
      • Between the hours 2 and 4 pm the average number of phone calls per minute coming into the switch board of a company is 2.5. find the probability that during one particular minute there will be
      • No phone call at all
      • Exactly two calls
      • At least Five calls
    • Home work
      • Find the probability that almost 5 defective fuses will be found in a box of 200 fuses. An experience shows that 2% of such fuses are defective.
    • Normal distribution
      • Both binomial and Poisson distributions consist of all the values (finite ) of a random variable that made up of these discrete and associated probabilities
      • Normal probability distribution is one of the most frequently used distribution. IT is normally described in terms of continuous curve in the shape of a bell (symmetrical)
    • Example
      • The weekly wages of 1,000 workmen are normally distributed around a mean of Rs. 70 and with a standard deviation of Rs.5 Estimate the number of workers whose weekly wages will be
      • Between Rs.70 and Rs.72
      • Between Rs.69 and Rs.72
      • More than Rs.75
      • Less than Rs. 63
    • Home work 1 Normal distribution
      • Consider a project that yields an average cash flow of Rs. 500 lakhs with a standard deviation of Rs. 60 lakhs. Calculate the following probabilities
        • Cash flow will be more than Rs. 560 lakhs
        • Cash flow will be less than Rs.420 lakhs
        • Cash flow will lie between Rs. 460 lakhs and Rs.540 lakhs
        • Cash flow will be more than Rs. 680 lakhs
    • Answers
      • 1.Z=(x-  )/  =(560-500)/60=1.0
      • 2.z=-1.33
      • 3.-.66 and .66(2*.2454=.4908
      • 4. 3.0, 0.0013