Short Solutions to Stat 230 Test 1, Jan 18, 2012

1. Four letters addressed to individuals W,X,Yand Zare randomly placed in four addressed envelopes,

one letter in each envelope.

[4] (a) List a 24-point sample space for this experiment. Be sure to explain your notation.

Suppose the four envelopes are arranged in a row as W XY Z. The 24 points can be listed below,

where the order of W,X,Yand Zrepresents the arrangements of four letters:

(W XY Z),(W XZY ),(W Y XZ),(W Y ZX),(W ZXY ),(W ZY X),

(XW Y Z),(XW ZX),(XY W Z),(XY ZW ),(XZW Y ),(XZY W ),

(Y W XZ),(Y W ZX),(Y ZW X),(Y ZXW ),(Y XW Z),(Y XZW ),

(ZW XY ),(ZW Y X),(ZY W X),(ZY XW ),(ZXW Y ),(ZXY W )

[3] (b) Assuming that the 24 sample points are equally likely, ﬁnd the probability of the following events:

A = “W’s letter goes into the correct envelope”

P(A) = 1×3!

24 =6

24 =1

4

B = “Exactly two letters go into the correct envelopes”

P(B) = (4

2)×1×1×1×1

24 =6

24 =1

4

C=“Exactly three letters go into the correct envelopes”

P(C) = 0

24 = 0