12W-Test1-Soln.pdf

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