This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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a) Let W denote a win, L denote a loss, and D denote a draw. Given probabilities for a single match: P(W) = 0.6 P(L) = 0.3 P(D) = 0.1 The team plays two matches. Here is the tree diagram: Match 1 Start with a single point. Branch 1: Win (W) with probability 0.6 Branch 2: Lose (L) with probability 0.3 Branch 3: Draw (D) with probability 0.1 Match 2 From each outcome of Match 1, draw three new branches for Match 2: If Match 1 was W: Win (W) with probability 0.6 (Path: WW) Lose (L) with probability 0.3 (Path: WL) Draw (D) with probability 0.1 (Path: WD) If Match 1 was L: Win (W) with probability 0.6 (Path: LW) Lose (L) with probability 0.3 (Path: LL) Draw (D) with probability 0.1 (Path: LD) If Match 1 was D: Win (W) with probability 0.6 (Path: DW) Lose (L) with probability 0.3 (Path: DL) Draw (D) with probability 0.1 (Path: DD) The possible outcomes and their probabilities are: P(WW) = 0.6 × 0.6 = 0.36 P(WL) = 0.6 × 0.3 = 0.18 P(WD) = 0.6 × 0.1 = 0.06 P(LW) = 0.3 × 0.6 = 0.18 P(LL) = 0.3 × 0.3 = 0.09 P(LD) = 0.3 × 0.1 = 0.03 P(DW) = 0.1 × 0.6 = 0.06 P(DL) = 0.1 × 0.3 = 0.03 P(DD) = 0.1 × 0.1 = 0.01 b) What is the probability that the team will: i) Win the two matches This corresponds to the outcome WW. P(Win two matches) = P(W) × P(W) = 0.6 × 0.6 = 0.36 The probability of winning two matches is 0.36. ii) Either wins all the matches or loses all the matches This means the outcomes WW or LL. These are mutually exclusive events. P(Wins all or loses all) = P(WW) + P(LL) P(WW) = 0.6 × 0.6 = 0.36 P(LL) = 0.3 × 0.3 = 0.09 P(Wins all or loses all) = 0.36 + 0.09 = 0.45 The probability of either winning all matches or losing all matches is 0.45. iii) Wins one match and loses one This means the outcomes WL or LW. These are mutually exclusive events. P(Wins one and loses one) = P(WL) + P(LW) P(WL) = P(W in match 1) × P(L in match 2) = 0.6 × 0.3 = 0.18 P(LW) = P(L in match 1) × P(W in match 2) = 0.3 × 0.6 = 0.18 P(Wins one and loses one) = 0.18 + 0.18 = 0.36 The probability of winning one match and losing one is 0.36. iv) Tie in one match This means exactly one match is a draw. The other match must not be a draw. Let P(not D) = P(W) + P(L) = 0.6 + 0.3 = 0.9. The possible outcomes are (D and not D) or (not D and D). P(Tie in one match) = P(D in match 1 and not D in match 2) + P(not D in match 1 and D in match 2) P(Tie in one match) = (P(D) × P(not D)) + (P(not D) × P(D)) P(Tie in one match) = (0.1 × 0.9) + (0.9 × 0.1) P(Tie in one match) = 0.09 + 0.09 = 0.18 Alternatively, this includes outcomes WD, LD, DW, DL: P(Tie in one match) = P(WD) + P(LD) + P(DW) + P(DL) P(Tie in one match) = (0.6 × 0.1) + (0.3 × 0.1) + (0.1 × 0.6) + (0.1 × 0.3) P(Tie in one match) = 0.06 + 0.03 + 0.06 + 0.03 = 0.18 The probability of a tie in one match is 0.18. 3 done, 2 left today. You're making progress.