Question

Three jars contain different chemicals R, S, and T. Their identifying labels were accidentally dropped during transportation. Suppose that a careless technician decides to put the labels on these jars at random without inspecting their contents.

a. List the sample space and assign probabilities to the elementary outcomes.

b. State the composition / elements of these events:

A = {there is exactly one match}

B = {all jars receive wrong labels}  

c.Find P( A ) and P( B ).

Answer 1

a.

Let the sample space be (X, Y) where X is the correct (before dropped) label and Y is the new label put on each jar.

The possible sample space are

[ {(R, R) , (S, S) , (T, T)}, {(R, R) , (S, T) , (T, S)}, {(R, S) , (S, R) , (T, T)}, {(R, S) , (S, T) , (T, R)}, {(R, T) , (S, S) , (T, R)},

{(R, T) , (S, R) , (T, S)}

As, each of the 6 outcomes have equal probabilities, the probabilities to the elementary outcomes = 1/6

b.

A = {there is exactly one match} , the sample spaces are,

{(R, R) , (S, T) , (T, S)}, {(R, S) , (S, R) , (T, T)}, {(R, T) , (S, S) , (T, R)},

B = {all jars receive wrong labels}  

{(R, S) , (S, T) , (T, R)} , {(R, T) , (S, R) , (T, S)}

c.

P(A) = n(A) / n = 3 / 6 = 1/2 = 0.5

P(A) = n(B) / n = 2 / 6 = 1/3  = 0.33