Biostatistics course Part 5 Binomial distribution Dr. Sc. Nicolas Padilla Raygoza Department of Nursing and Obstetrics Division of Health Sciences and Engineering Campus Celaya Salvatierra Universidad de Guanajuato Mexico Lecture updated on November 17, 2012 All Supercourse biostatistics lectures available here - http://www.pitt.edu/~super1/ResearchMethods/index1.htm

Biosketch Medical Doctor by University Autonomous of Guadalajara. Pediatrician by the Mexican Council of Certification on Pediatrics. Postgraduate Diploma on Epidemiology, London School of Hygine and Tropical Medicine, University of London. Master Sciences with aim in Epidemiology, Atlantic International University. Doctorate Sciences with aim in Epidemiology, Atlantic International University. Professor Titular A, Full Time, University of Guanajuato. Level 1 National Researcher System padillawarm@gmail.com padillawarm@gmail.com

Competencies The reader will define what is binomial distribution. He (she) will know how is binomial distribution.

Introduction We know how calculate single probabilities, but now, we have to calculate more complex probabilities. Example 100 new born in a maternity in Celaya. 55 were females and 45 were males. Probability to be girl was 55/100 = 0.55 Probability to be boy, was 45/100=0.45 What is the probability of two males in the next three new borns in this maternity?

Introduction Two males between three new borns can occur: Male Male Female (MMF) Male Female Male (MFM) Female Male Male (FMM) A, B and C are mutually excluded, and Probability (HHM) + Probability (HMH) + Probability (MHH) First, we need to know the probability of each combination: A) 0.45 x 0.45 x 0.55 = 0.13613 = 13.61% B) 0.45 x 0.55 x 0.45 = 0.13613 = 13.61% C) 0.55 x 0.45 x 0.45 = 0.13613 = 13.61% Then, the probability of two males in the next three new born is: 0.1361 + 0.1361 + 0.1361 = 0.4083 = 40.83%

Introduction What is the probability that at least 1 of the next three new born be male? The combinations: MFF, FMF, FFM, MMF, MFM, MMF, MMM. To calculate probability in each combination and then add them, consume time. The possible combinations of gender in three new born are 8: MFF, FMF, FFM, MMF, MFM, MMF, MMM, FFF. If we calculate the probability of three females (FFF) should be: 0.55 x 0.55 x 0.55 =0.1664 = 16.64%. If to 1 (total of combinations), we subtract 0.16644, shall obtain the probability of al least 1 male in the next three new born = 1-0.1664 = 0.8336 = 83.36%.

In anyone calculation of probability, we should count how many combinations of an event will produce an result; to calculate the probability of each combination and then add the probability of all combinations, because they are mutually excluded.

Binomial distribution Describe probability of a characteristic that only can take two values. This chart describe the distribution of probability of number of males in a group of ten person. It is the binomial distribution that describe the characteristic of a variable, that it can take only two values. The high of each bar is less than 11 and the add of all bars is = 1. When the more high probability is near to the middle of range, the distribution is symmetrical. When the higher probability is near to the tails the distribution is skewed. It is skewed to the right if the higher probability is near 0 and it is skewed to the left when the higher probability is near 10 (in this example).

Bibliografía 1.- Last JM. A dictionary of epidemiology. New York, 4ª ed. Oxford University Press, 2001:173. 2.- Kirkwood BR. Essentials of medical ststistics. Oxford, Blackwell Science, 1988: 1-4. 3.- Altman DG. Practical statistics for medical research. Boca Ratón, Chapman & Hall/ CRC; 1991: 1-9.