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Review: INDEPENDENCE OF EVENTS

• TWO EVENTS A AND B ARE SAID TO BE INDEPENDENT IF ANY OF THE FOLLOWING EQUIVALENT CONDITIONS ARE TRUE:

)()( APBAP

)()( BPABP

)().()( BPAPBAP

Chapter 12: Joint DistributionsData classifying 356 male federal employees

based on socio-economic status and smoking

Socio-economic Status

Smoking High Middle Low

Current 51 22 43

Former 92 21 28

Never 68 9 22

Chapter 12: Joint Distributions: Intro to Modeling

Variables of InterestX: Smoking and Y: Socio-economic status

X = 0 mean person is current smokerX = 1 mean person is former smokerX = 2 mean person has never smokedY = 0 mean person has high socio-economic statusY = 1 mean person has medium socio-economic statusY = 2 mean person has low socio-economic status

Chapter 12: Joint DistributionsJoint probability distribution for smoking andsocio-economic status.

Socio-economic Status (Y)

Smoking (X) High (0) Middle (1) Low (2)

Current (0) .143 .062 .121

Former (1) .258 .059 .079

Never (2) .191 .025 .062

Chapter 12: Joint DistributionsMarginal Distribution for Smoking

Marginal Distribution for socio-economic status

X 0 1 2

P(X=u) 0.326 0.396 .278

Y 0 1 2

P(Y =v) 0.592 0.146 0.262

Chapter 12: Independence and Joint Distributions

Independence from a joint distribution table:Check: P(X=u , Y=v) = P(X=u)P(Y=v)

Note: P(X=u , Y=v) means P(X=u and Y=v). Example:

P(X=1, Y=2) = 0.079P(X=1) = 0.396 , P(Y=2)= 0.262 P(X=1)P(Y=2) = 0.396*0.262 = 0.103752. This suffices to conclude that smoking and socio-

economic status are not independent.

Chapter 12: Sampling without Replacement

In a drawer there are has 6 black and 8 blue socks. Select two socksrandomly.

What is the probability that you get two socks that are of different color?

What is the probability that the first sock is black and thesecond is blue?

Second Sock =Black Second Sock = Blue

First Sock = Black 30/182 48/182

First Sock = Blue 48/182 56/182

Chapter 13: Hyper Geometric Distributions

Modeling “Sampling without Replacement Problems”

Example: Suppose that you have a bag filled with 50marbles, 15 of which are green. What is the probability ofchoosing exactly 3 green marbles if a total of 10 marblesare selected?

Chapter 13: Hyper Geometric Distributions

You are president of an on-campus special eventsorganization. You need a committee of 7 to plan aspecial birthday party for the president of thecollege. Your organization consists of 18 womenand 15 men. What is the probability that yourcommittee will have 4 female?