Short Solutions to Stat 230 Test 1, Jan 18, 2012
1. Four letters addressed to individuals W, X, Y and Z are 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 WXY Z. The 24 points can be listed below,
where the order of W, X, Y and Z represents the arrangements of four letters:
(WXY Z); (WXZY ); (WY XZ); (WY ZX); (WZXY ); (WZY X);
(XWY Z); (XWZX); (XY WZ); (XY ZW); (XZWY ); (XZY W);
(Y WXZ); (Y WZX); (Y ZWX); (Y ZXW); (Y XWZ); (Y XZW);
(ZWXY ); (ZWY X); (ZY WX); (ZY XW); (ZXWY ); (ZXY W)
[3] (b) Assuming that the 24 sample points are equally likely, ▯nd the probability of the following events:
A = \W’s letter goes into the correct envelope"
1 ▯ 3! 6 1
P(A) = 24 = 24= 4
B = \Exactly two letters go into the correct envelopes"
4
P(B) = (2) ▯ 1 ▯ 1 ▯ 1 ▯=1 6 = 1
24 24 4
C=\Exactly three letters go into the correct envelopes"
0
P(C) = 24 = 0 2. In Lotto 6/49 you purchase a lottery ticket with 6 di▯erent numbers, selected from the set f1;2;▯▯▯;49g.
In the draw, six (di▯erent) numbers are randomly selected from the same set. Note that the selection
of numbers is without replacement. Find the probability of the following events:
[2] (a) A = \Your ticket has all the 6 numbers which are drawn (you win the main Jackpot)"
! !
49 6
There are equally likely outcomes and there is only = 1 way to match all 6 numbers
6 6
drawn. Therefore,
1
P(A) = ▯49
6
Note: P(A) = 1=13983816, in other words, the probability of winning a Jackpot is roughly one
out of 14 million.
[2] (b) B = \Your ticket matches exactly 5 of the 6 numbers drawn"
From the 6 drawn numbers, you must choose 5. From the 43 non-drawn numbers, choose 1.
Therefore, ▯ ▯▯ ▯
6 43

More
Less