Binomial distribution for mathematics pre u

Binomial distribution

1. Chapter 9 Special Probability Distribution

2. Chapter Goals After completing this chapter, you should be able to: Identify probability problems with binomial distributions Describe and compute probabilities for a Binomial distributions Identify probability problems with Poisson distribution Describe and compute probabilities for a P oisson distributions Find mean and variance for Binomial and Poisson Distributions

3. Chapter Goals After completing this chapter, you should be able to: Identify probability problems with normal distribution Describe the characteristics normal distributions Translate normal distribution problems into standardized normal distribution problems Find probabilities using a normal distribution table (continued)

4. Probability Distributions Continuous Probability Distributions Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Uniform Normal Exponential

5. The Binomial Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions 4.4

6. Binomial Distribution Consider an experiment with only two outcomes: “ success ” or “ failure ” - The Bernoulli Trial Let p denote the probability of success Let q be the probability of failure and q =1 – p If there are n number independent Bernoulli Trials And let random variable X= no. of success X has a Binomial R.V. i.e it has a Binomial p.d.f .   is the no. of trials the prob. of success in each trial  

7. Binomial Distribution Formula   = probability of successes in trials, with probability of success on each trial = number of ‘successes’ in sample, ( = 0, 1, 2, ..., ) = sample size (number of trials or observations) = probability of “success”   Example: Flip a coin four times, let x = # heads: = 4 = 0.5 1 - = (1 - 0.5) = 0.5 x = 0, 1, 2, 3, 4  

8. Sequences of x Successes in n Trials The number of sequences with x successes in n independent trials is : Where n! = n · (n – 1) · (n – 2) · . . . · 1 and 0! = 1 These sequences are mutually exclusive, since no two can occur at the same time

9. Example: Calculating a Binomial Probability What is the probability of one success in five observations if the probability of success is 0.1 ? x = 1, n = 5, and p = 0.1

10. Example: Calculating a Binomial Probability A fair dice is rolled six times. Let be the number of times the outcome is a multiple of 3.   Find the probability distribution function of   Success = getting no. multiple of 3 = 3 or 6

11. Example: Calculating a Binomial Probability A fair dice is rolled six times. Let be the number of times the outcome is a multiple of 3.   Find the probability distribution function of   1 2 3 4 5 6 0.0878 0.2634 0.3292 0.2195 0.0823 0.0165 0.0014 1 2 3 4 5 6 0.0878 0.2634 0.3292 0.2195 0.0823 0.0165 0.0014

12. Using Binomial Tables We can use statistical table for Binomial Probability distribution. However, the table gives the cumulative probability for a given values of and   The table is given on page 307-312 in the text book

13. Using Binomial Tables N x … p=.20 p=.25 p=.30 p=.35 p=.40 p=.45 p=.50 10 1 2 3 4 5 6 7 8 9 10 … … … … … … … … … … … 1.0000 0.8929 0.6242 0.3222 0.1209 0.0328 0.0064 0.0009 0.0001 0.0000 0.0000 1.0000 0.9437 0.7560 0.4744 0.2241 0.0781 0.0197 0.0035 0.0004 0.0000 0.0000 1.0000 0.9718 0.8607 0.6172 0.3504 0.1503 0.0473 0.0106 0.0016 0.0001 0.0000 1.0000 0.9865 0.9140 0.7384 0.4862 0.2485 0.0949 0.0260 0.0048 0.0005 0.0000 1.0000 0.9940 0.9536 0.8327 0.6177 0.3669 0.1662 0.0548 0.0123 0.0017 0.0001 1.0000 0.9975 0.9767 0.9004 0.7340 0.4956 0.2616 0.1020 0.0274 0.0045 0.0003 1.0000 0.9990 0.9893 0.9453 0.8281 0.6230 0.3770 0.1719 0.0547 0.0107 0.0010 Examples: n = 10, x = 3, p = 0.35: P(x 3) = 0.7384 n = 10, x = 8, p = 0.45: P(x 8) = 0.0274  

14. Example: Calculating a Binomial Probability Let be a random variable with =10 and =0.4, that is Find the following probabilities:   a)   = 0.6177 (read direct from the table) b)   =   =   c)   =   =   d )   =   =   e )   =   =   =  

15. Possible Binomial Distribution Settings A manufacturing plant labels items as either defective or acceptable A firm bidding for contracts will either get a contract or not A marketing research firm receives survey responses of “yes I will buy” or “no I will not” New job applicants either accept the offer or reject it

16. Binomial Distribution The shape of the binomial distribution depends on the values of p and n n = 5 p = 0.1 n = 5 P = 0.5 Mean .2 .4 .6 1 2 3 4 5 x P(x) .2 .4 .6 1 2 3 4 5 x P(x) Here, n = 5 and p = 0.1 Here, n = 5 and p = 0.5

17. Binomial Distribution Mean and Variance Mean Variance and Standard Deviation Where n = sample size p = probability of success q=(1 – p) = probability of failure

18. …..to be continued…. Q and A