Question

A certain honest wheel of chance is divided into three equal segments colored green (G), red (R), and yellow (Y), respectively. You spin the wheel twice and take the outcome to be the resulting color sequence—GR, RG, and so forth. Let A = “neither color is yellow” and let B = “matching colors.” Draw the Venn diagram and calculate P(A), P(B), P(AB), and P(A + B).

Answer 1

Solution :

Sample Space of the outcome of spinning a certain honest wheel twice :

{RR,GR,YR,RG,GG,YG,RY,GY,YY} = 9 outcomes

where, R= Red, G= Green and Y= Yellow.

Probability of an event is given by the formula :

Here, E= Event in consideration

The Venn Diagram is drawn as below :

From the diagram,

A= elements where neither color is yellow=4,

B = elements with matching colors=3,

w= Total elements,

= All elements belonging to A and B

(i) Probability of Event A :

Since, A = neither color is yellow = {RR,GR,RG,GG}= 4

Total number of favorable outcomes= {RR,GR,YR,RG,GG,YG,RY,GY and YY}=9

Therefore, probability, P(A)=

(ii) Probability of Event B :

Since, B= elements with matching colors= {RR,GG,YY}=3

Total number of favorable outcomes={RR,GR,YR,RG,GG,YG,RY,GY and YY}=9

Therefore, probability, P(B) =

(iii) Calculation of P(AB):

We know that,

Where, = All elements present in A and B put together without repeating = {RR,GR,RG,GG,YY}=5

Therefore,  

(iv) Calculation of P(A)+ P(B):

From (i) and (ii),