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# All Probability

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### All Probability

1. 1. Questions
2. 2. Question 5 <ul><li>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. </li></ul><ul><li>41/90 </li></ul>
3. 3. Questions Random variable
4. 4. Questions <ul><li>Find the expected value of the heads when two coins are tossed </li></ul>
5. 5. Question no 2 <ul><li>A box contains six tickets. Two of the tickets carry a price of Rs. 5 each and the other four prices of Re.1 </li></ul><ul><li>A) If one ticket is drawn what is the expected value of the price? </li></ul><ul><li>B)If two tickets are drawn ,what is the expected value of the price. </li></ul>
6. 6. Questions Binomial Distribution
7. 7. The binomial distribution describes the behavior of a count variable X if the following conditions apply: <ul><ul><ul><li>1: The number of observations n is fixed. </li></ul></ul></ul><ul><ul><ul><li>2: Each observation is independent. </li></ul></ul></ul><ul><ul><ul><li>3: Each observation represents one of two outcomes (&quot;success&quot; or &quot;failure&quot;). </li></ul></ul></ul><ul><ul><ul><li>4: The probability of &quot;success&quot; p is the same for each outcome. </li></ul></ul></ul><ul><ul><li>If these conditions are met, then X has a binomial distribution with parameters n and p, abbreviated B(n,p). </li></ul></ul>
8. 8. <ul><li>A random variable X is said to follow Binomial distribution with parameters n and p if its probability function is </li></ul><ul><li>f(x)= nC x p x q n-x </li></ul><ul><li>Where x= 0,1,2,….n </li></ul><ul><li>P + q = 1 </li></ul>
9. 9. Mean of binomial distribution <ul><li>Mean of binomial distribution = np </li></ul><ul><li>n- number of trials </li></ul><ul><li>p- probability of success </li></ul>
10. 10. Standard deviation of a binomial distribution <ul><li>Standard deviation of a binomial distribution = npq </li></ul><ul><li>n- number of trials </li></ul><ul><li>p– probability of success </li></ul><ul><li>q- Probability Of Failure or (1-p) </li></ul>
11. 11. Question no 1 <ul><li>Four coins are tossed simultaneously. What is the probability of getting two heads </li></ul>
12. 12. Question No 2 <ul><li>Eight unbiased coins are tossed simultaneously. Find the probability of getting </li></ul><ul><li>Exactly four heads </li></ul><ul><li>No heads at all </li></ul><ul><li>6 or more heads </li></ul><ul><li>Utmost two heads </li></ul><ul><li>Number of heads ranging from 3 to 5 </li></ul>
13. 13. Question no 3 <ul><li>Eight coins are tossed simultaneously 256 times . Find the expected frequencies </li></ul><ul><li>Find mean and Standard Deviation </li></ul>
14. 14. Home work <ul><li>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 </li></ul>
15. 15. Poison Distribution Called law of improbable events-describe the behaviour of rare events Discrete Probability distribution
16. 16. Formula <ul><li>The Probability of N success out of n trials is given by </li></ul><ul><li>Where x is a discrete random variable assuming values 0,1,2… </li></ul><ul><li>m is called parameter of Poisson distribution </li></ul>
17. 17. Example <ul><li>., Find the probability of 4 customers arriving in 3 minutes when the mean is 3.6. </li></ul>
18. 18. Question no 1 <ul><li>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 </li></ul>
19. 19. Question No 2 <ul><li>Fit a poison distribution to the following data and calculate the theoretical frequencies </li></ul><ul><li>X 0 1 2 3 4 </li></ul><ul><li>Y 123 59 14 3 1 </li></ul>
20. 20. Home Work <ul><li>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 </li></ul><ul><li>No phone call at all </li></ul><ul><li>Exactly two calls </li></ul><ul><li>At least Five calls </li></ul>
21. 21. Home work <ul><li>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. </li></ul>
22. 22. Normal distribution <ul><li>Both binomial and Poisson distributions consist of all the values (finite ) of a random variable that made up of these discrete and associated probabilities </li></ul><ul><li>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) </li></ul>
23. 23. Example <ul><li>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 </li></ul><ul><li>Between Rs.70 and Rs.72 </li></ul><ul><li>Between Rs.69 and Rs.72 </li></ul><ul><li>More than Rs.75 </li></ul><ul><li>Less than Rs. 63 </li></ul>
24. 24. Home work 1 Normal distribution <ul><li>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 </li></ul><ul><ul><li>Cash flow will be more than Rs. 560 lakhs </li></ul></ul><ul><ul><li>Cash flow will be less than Rs.420 lakhs </li></ul></ul><ul><ul><li>Cash flow will lie between Rs. 460 lakhs and Rs.540 lakhs </li></ul></ul><ul><ul><li>Cash flow will be more than Rs. 680 lakhs </li></ul></ul>
25. 25. Answers <ul><li>1.Z=(x-  )/  =(560-500)/60=1.0 </li></ul><ul><li>2.z=-1.33 </li></ul><ul><li>3.-.66 and .66(2*.2454=.4908 </li></ul><ul><li>4. 3.0, 0.0013 </li></ul>