chapter 15

Chapter 15

Week 5, Friday

Population Cross-tabs

Ohio Texas New York (total)

Like the Song 200 100 5 305

Don’t Like the Song 500 125 70 695

(total) 700 225 75 1000

Suppose there are 1000 employees invited to a company party. The company has locations in Ohio, Texas, and New York. Corporate is considering the song “My Kind of Party” by Brantley Gilbert for the Welcome Video and wants to know how the 1000 employees in attendance feel about this song. The cross-tab below describes the 1000 attending employees’ feelings toward this song.

Why is this not a sample?If I randomly choose 1 person from the company, what is the likelihood that the employee will like the song? 305/1000 = 30.5%If I randomly choose 1 person from Texas, what is the likelihood that the employee will like the song? 100/225 = 44.4%

Population Cross-tabs

Ohio Texas New York (total)

Like the Song 200 100 5 305

Don’t Like the Song 500 125 70 695

(total) 700 225 75 1000

Suppose there are 1000 employees invited to a company party. The company has locations in Ohio, Texas, and New York. Corporate is considering the song “My Kind of Party” by Brantley Gilbert for the Welcome Video and wants to know how the 1000 employees in attendance feel about this song. The cross-tab below describes the 1000 attending employees’ feelings toward this song.

If I randomly choose 1 person from Texas, what is the likelihood that the employee will like the song? 100/225 = 44.4%

P[Like the song | Texas] = = P[Like the song AND Texas] /

P[Texas] = (100/1000) / (225/1000) = 100/225

Population Cross-tabs

Ohio Texas New York (total)

Like the Song 200 100 5 305

Don’t Like the Song 500 125 70 695

(total) 700 225 75 1000

Suppose there are 1000 employees invited to a company party. The company has locations in Ohio, Texas, and New York. Corporate is considering the song “My Kind of Party” by Brantley Gilbert for the Welcome Video and wants to know how the 1000 employees in attendance feel about this song. The cross-tab below describes the 1000 attending employees’ feelings toward this song.

If I randomly choose 1 person from Texas, what is the likelihood that the employee will like the song? 100/225 = 44.4%

The general formula for events A and B:

P[ A | B ] = P[ A and B ] / P[B]

Recall the Coin Example

Consider two fair coins:

Tails, Tails

Tails, Heads

P[ at least one heads ] = 3/4P[ two heads ] = 1/4P[ at least one heads AND two heads ] =1/4

Heads, Tails

Heads, Heads

P[ two heads | At least one heads ] = ?= P[ 2 heads and At least one heads ] / P[at least one heads]= (1/4) / (3/4) = 1/3 = 33%

Definition: Mutually ExclusiveTwo events are mutually exclusive if they have nothing in common.

That is P[ A and B ] = 0

Definition: Mutually Exclusive

Consider two fair coins:

Tails, Tails

Tails, Heads

Consider the events:

1 heads and 1 tails

2 heads

They are mutually exclusive

Heads, Tails

Heads, Heads

Definition: Mutually Exclusive

Consider two fair coins:

Tails, Tails

Tails, Heads

As a result:

(1) P[ {HT,TH} and HH ] = 0

(2) P[ {HT,TH} or HH ]

= P[HT, TH] + P[HH] - P[{HT,TH} and HH]

= P[HT, TH] + P[HH]

Heads, Tails

Heads, Heads

Definition: Mutually Exclusive

Consider two fair coins:

Tails, Tails

Tails, Heads

General Equation:

For Mutually Exclusive events, A and B:

P[ A or B ] = P[A] + P[B]

Heads, Tails

Heads, Heads

Definition: Independent

Two events are independent if the outcome of one event does not affect the outcome of the other.

That is P[ A | B ] = P[A]

Also: P[A and B] = P[A] * P[B]

Example 1

Example: Flip three coins. Suppose these coins are independent of each other. Calculate the probability of flipping three heads.

Solution: Let H1 be the event that the first flip is heads

Let H2 be the event that the second flip is heads

Let H3 be the event that the third flip is heads

P[H1 and H2 and H3] = P[H1]*P[H2]*P[H3]

= (1/2)*(1/2)*(1/2) = 1/8

Example 2 (chapter 14 #32B)

Example: Consider the blood types:P[type O] = 45%, P[type B] = 11%

P[type A] = 40%, P[type AB] = 4%

Consider four independent donors

Question 1:

P[ all are type O ]

= P[p1 is O and p2 is O and p3 is O and p4 is O]

= P[p1 is O]*P[p2 is O]*P[p3 is O]*P[p4 is O]

= 45% * 45% * 45% * 45%

= 4.1%

Example 2 (chapter 14 #32B)

Example: Consider the blood types:P[type O] = 45%, P[type B] = 11%

P[type A] = 40%, P[type AB] = 4%

Consider four independent donors

Question 2:

P[ no one is AB ]

= P[p1 not AB and p2 not AB and p3 not AB and p4 not AB]

= P[p1 not AB]*P[p2 not AB]*P[p3 not AB]*P[p4 not AB]

= 96% * 96% * 96% * 96%

= 84.9%

Example 2 (chapter 14 #32B)

Example: Consider the blood types:P[type O] = 45%, P[type B] = 11%

P[type A] = 40%, P[type AB] = 4%

Consider four independent donors

Question 3:

P[ not everyone is type A ]

= 1 – P[ everyone is type A]

= 1 - P[p1 is A and p2 is A and p3 is A and p4 is A]

= 1 - P[p1 is A]*P[p2 is A]*P[p3 is A]*P[p4 is A]

= 100% - (40% * 40% * 40% * 40%)

= 97.4%

Example 2 (chapter 14 #32B)

Example: Consider the blood types:P[type O] = 45%, P[type B] = 11%

P[type A] = 40%, P[type AB] = 4%

Consider four independent donors

Question 4:

P[ at least one person is type B ]

= 1 – P[ nobody is type B ]

= 1 - P[p1 not B and p2 not B and p3 not B and p4 not B]

= 1 - P[p1 not B]*P[p2 not B]*P[p3 not B]*P[p4 not B]

= 100% - (89% * 89% * 89% * 89%)

= 37.3%

Example 3

Answer:

P[ like the song ] = 305/1000 = 30.5%

P[ like the song | from texas ] = 100/225 = 44.44%

Since 44.44% is not equal to 30.5%,

these events are NOT independent.

Question: Are “liking the song” and “being from Texas” independent events?

Ohio Texas New York (total)

Like the Song 200 100 5 305

Don’t Like the Song 500 125 70 695

(total) 700 225 75 1000

Example 4: A different song

Answer:

P[ like the song ] = 200/1000 = 20%

P[ like the song | from ohio ] = 100/500 = 20%

Since both values are 20%,

these events ARE independent.

Question: Are “liking the song” and “being from Ohio” independent events?

Ohio Texas New York (total)

Like the Song 100 195 5 200

Don’t Like the Song 400 330 70 800

(total) 500 425 75 1000

Example 4: A different song

Alternative Answer:

P[ from ohio ] = 500/1000 = 50%

P[ from ohio | like the song ] = 100/200 = 50%

Since both values are 50%,

these events ARE independent.

Question: Are “liking the song” and “being from Ohio” independent events?

Ohio Texas New York (total)

Like the Song 100 195 5 200

Don’t Like the Song 400 330 70 800

(total) 500 425 75 1000