Construct and Interpret Binomial Distributions

Transcription

1 CH 6.2 Distribution.notebook A random variable is a variable whose values are determined by the outcome of the experiment. 1

2 CH 6.2 Distribution.notebook A probability distribution is a function which maps each value of a random variable onto its probability. 2

3 CH 6.2 Distribution.notebook The probability distribution below comes from the sum of rolling two six sided dice. Sum Probability

4 CH 6.2 Distribution.notebook If you add all of the probabilities, the sum is =1 4

5 CH 6.2 Distribution.notebook If a family had three children and assume that the births of boys and girls is equally likely. Let the random variable of the distribution stand for the number of boys. The domain of the random variable is: 0, 1, 2, 3 5

6 CH 6.2 Distribution.notebook There are 2 3 or 8 possible outcomes: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG The probability for each value of the random variable is: x = number of boys P(x) 6

7 CH 6.2 Distribution.notebook A binomial experiment meets the following criteria: There are n independent trials Each trial has only two possible outcomes: success or failure The probability of success is the same for each trial. This probability is denoted by p. The probability of failure, q, is q = 1 p 7

8 CH 6.2 Distribution.notebook For a binomial experiment, the probability of exactly k successes in n trials is: 8

9 CH 6.2 Distribution.notebook Assume a basketball player makes 75% of free throw attempts and that the attempts are independent of each other. What is the probability that the player makes exactly 5 free throws in 8 attempts. p=0.75 n = 8 q = = 0.25 r = 5 9

10 CH 6.2 Distribution.notebook 10

11 CH 6.2 Distribution.notebook Now, assume a basketball player makes 75% of free throw attempts and that the attempts are independent of each other. Determine the probability distribution for the number of free throws made in 8 attempts. 11

12 CH 6.2 Distribution.notebook ( n C k )(p k )(q n-k ) n=8, p = 0.75, q=1-0.75=0.25 P(x)=( 8 C x )(0.75) x (0.25) 8-x 12

13 CH 6.2 Distribution.notebook x P(x) 0 ( 8 C 0 )*(0.75) 0 *(0.25) ( 8 C 1 )*(0.75) 1 *(0.25) ( 8 C 2 )*(0.75) 2 *(0.25) ( 8 C 3 )*(0.75) 3 *(0.25) ( 8 C 4 )*(0.75) 4 *(0.25) ( 8 C 5 )*(0.75) 5 *(0.25) ( 8 C 6 )*(0.75) 6 *(0.25) ( 8 C 7 )*(0.75) 7 *(0.25) ( 8 C 8 )*(0.75) 8 *(0.25)

14 CH 6.2 Distribution.notebook P(x)=( 8 C x )(0.75) x (0.25) 8-x 14

15 CH 6.2 Distribution.notebook From the previous problem, what is the probability of making 6 or more free throws? ( 8 C 6 )*(0.75) 6 *(0.25) ( 8 C 7 )*(0.75) 7 *(0.25) ( 8 C 8 )*(0.75) 8 *(0.25)

16 CH 6.2 Distribution.notebook As the probability of success changes, the shape of the graph of the probably distribution changes. 16

17 CH 6.2 Distribution.notebook A distribution that can be divided into two equal parts is called symmetric. 17

18 CH 6.2 Distribution.notebook A distribution that is not symmetric is skewed. 18

19 Attachments CH 6.2 Binomial Distribution Graphs.xls