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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You're on a roll — Question 22: a) Draw a tree diagram to illustrate all the possible outcomes and associated probabilities. Step 1: Determine the probability of biscuits made by each machine. • Machine K makes 45% of biscuits: P(K) = 0.45 • Machine J makes 30% of biscuits: P(J) = 0.30 • Machine L makes the rest: P(L) = 1 - P(K) - P(J) = 1 - 0.45 - 0.30 = 1 - 0.75 = 0.25 Step 2: Determine the conditional probabilities of a biscuit being broken (B) or not broken (B') for each machine. • For Machine J: P(B|J) = 0.02, so P(B'|J) = 1 - 0.02 = 0.98 • For Machine K: P(B|K) = 0.03, so P(B'|K) = 1 - 0.03 = 0.97 • For Machine L: P(B|L) = 0.05, so P(B'|L) = 1 - 0.05 = 0.95 Step 3: Draw the tree diagram. ` Start | ----------------------------------------------------- | | | K (0.45) J (0.30) L (0.25) | | | ----------------- ----------------- ----------------- | | | | | | B (0.03) B'(0.97) B (0.02) B'(0.98) B (0.05) B'(0.95) | | | | | | P(K and B) P(K and B') P(J and B) P(J and B') P(L and B) P(L and B') = 0.450.03 = 0.450.97 = 0.300.02 = 0.300.98 = 0.250.05 = 0.250.95 = 0.0135 = 0.4365 = 0.0060 = 0.2940 = 0.0125 = 0.2375 ` The possible outcomes and their associated probabilities are: • P(Machine K and Broken) = 0.45 × 0.03 = 0.0135 • P(Machine K and Not Broken) = 0.45 × 0.97 = 0.4365 • P(Machine J and Broken) = 0.30 × 0.02 = 0.0060 • P(Machine J and Not Broken) = 0.30 × 0.98 = 0.2940 • P(Machine L and Broken) = 0.25 × 0.05 = 0.0125 • P(Machine L and Not Broken) = 0.25 × 0.95 = 0.2375 What's next?