Question

A researcher at a large university wanted to investigate if a student's seat preference was related in any way to the gender of the student. The researcher divided the lecture room into three sections (1-front, middle of the room, 2-front, sides of the classroom, and 3-back of the classroom, both middle and sides) and noted where his students sat on a particular day of the class. The researcher's summary table is provided below.

Area (1) Area (2) Area (3) | Total Males Females Total 14 39 31 23 72

(a) A person is randomly selected. Find the probability that the person is male or sits in the front, middle of the room.

(b) Suppose a person sitting in the front, middle portion of the class is randomly selected to answer a question. Find the probability the person selected is a female.

(c) Are the events ‘person is male’ and ‘sits in the front’ middle independent? Give reasons for your answer.   

Answer 1

Solution:

Given:

Area (1) Area (2) Area (3) | Total Males Females Total 14 39 31 23 72

Part (a) A person is randomly selected. Find the probability that the person is male or sits in the front, middle of the room.

P( male or sits in the front, middle of the room ) =..........?

Let M = Male and we have Area(a) = sits in the front, middle of the room

Thus

P( M or Area(1) ) =..........?

P(M or Area(1) ) = P(M) + P( Area(1) ) - P( M and Area(1) )

P(M or Area(1) ) = 33/72 + 31/72 - 17/72

P(M or Area(1) ) = ( 33+31 - 17) / 72

P(M or Area(1) ) = 47 / 72

P(M or Area(1) ) = 0.6527778

P(M or Area(1) ) = 0.6528

Part (b) Suppose a person sitting in the front, middle portion of the class is randomly selected to answer a question. Find the probability the person selected is a female.

That is find:

P( Female | person sitting in the front, middle portion ) =.........?

P( F | Area(1) ) = .................?

Using conditional probability we get:

P(F and Area(1)) P(F|Area(1)) = ? P(Area(1)

P(F|Area(1)) = 14/72 39/72

P(F|Area(1)) = 11

P(FArea 1)) = 0.358974

P(FArea 1)) = 0.3590

Part (c) Are the events ‘person is male’ and ‘sits in the front’ middle independent? Give reasons for your answer.   

If events A and B are independent if and only if:

P( A and B) = P( A ) X P(B)

We have: events person is male’ and ‘sits in the front’ middle

We have:

P( M and Area(1) ) = 17/72

P( M and Area(1) ) = 0.2361

Now find

P(M) X P(Area(1) ) = ..............?

P(M) X P(Area(1) ) = ( 33/72 ) X ( 31/72)

P(M) X P(Area(1) ) = ( 0.4583333 ) X ( 0.4305556)

P(M) X P(Area(1) ) = 0.1973380

P(M) X P(Area(1) ) = 0.1973

Since P( M and Area(1) ) is not equal to P(M) X P(Area(1) ), the events ‘person is male’ and ‘sits in the front’ middle are not independent