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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Here are the solutions to the questions. 8 (i) Given P(A) = (2)/(5) and P(A B) = (7)/(10). The general formula for the union of two events is P(A B) = P(A) + P(B) - P(A B). a) Find P(B), if A and B are mutually exclusive. Step 1: For mutually exclusive events, P(A B) = 0. The formula simplifies to P(A B) = P(A) + P(B). Step 2: Substitute the given values into the formula. (7)/(10) = (2)/(5) + P(B) Step 3: Solve for P(B). P(B) = (7)/(10) - (2)/(5) P(B) = (7)/(10) - (4)/(10) P(B) = (3)/(10) The probability P(B) when A and B are mutually exclusive is (3)/(10). b) Find P(B), if A and B are independent. Step 1: For independent events, P(A B) = P(A) · P(B). Substitute this into the general formula: P(A B) = P(A) + P(B) - P(A) · P(B). Step 2: Substitute the given values into the formula. (7)/(10) = (2)/(5) + P(B) - ((2)/(5)) P(B) Step 3: Solve for P(B). (7)/(10) - (2)/(5) = P(B) - (2)/(5) P(B) (7)/(10) - (4)/(10) = P(B) (1 - (2)/(5)) (3)/(10) = P(B) ((5)/(5) - (2)/(5)) (3)/(10) = P(B) ((3)/(5)) P(B) = (3)/(10) ÷ (3)/(5) P(B) = (3)/(10) × (5)/(3) P(B) = (15)/(30) P(B) = (1)/(2) The probability P(B) when A and B are independent is (1)/(2). 8 (ii) Let X, Y, Z be the events that a truck is hired from Company X, Y, or Z, respectively. Let L be the event that a truck arrives late. Given probabilities: P(X) = 40\% = 0.40 P(Y) = 50\% = 0.50 P(Z) = 10\% = 0.10 Given conditional probabilities of arriving late: P(L|X) = 9\% = 0.09 P(L|Y) = 6\% = 0.06 P(L|Z) = 20\% = 0.20 c) Calculate the probability that the truck will arrive late, P(L). Step 1: Use the law of total probability. P(L) = P(L|X)P(X) + P(L|Y)P(Y) + P(L|Z)P(Z) Step 2: Substitute the given values. P(L) = (0.09)(0.40) + (0.06)(0.50) + (0.20)(0.10) Step 3: Calculate the products. P(L) = 0.036 + 0.030 + 0.020 Step 4: Sum the probabilities. P(L) = 0.086 The probability that the truck will arrive late is 0.086. d) Calculate the probability that the truck will not arrive late, P(L'). Step 1: The probability of an event not happening is 1 minus the probability of it happening. P(L') = 1 - P(L) Step 2: Substitute the value of P(L) from part (c). P(L') = 1 - 0.086 Step 3: Calculate the result. P(L') = 0.914 The probability that the truck will not arrive late is 0.914. Given that a truck hired arrives late, find the probability that it comes from Y. Step 1: We need to find the conditional probability P(Y|L). Use Bayes' theorem: P(Y|L) = (P(L|Y)P(Y))/(P(L)) Step 2: Substitute the known values. P(L) was calculated in part (c). P(Y|L) = ((0.06)(0.50))/(0.086) Step 3: Calculate the numerator. P(Y|L) = (0.030)/(0.086) Step 4: Calculate the final probability. P(Y|L) ≈ 0.348837 P(Y|L) ≈ 0.349 (to 3 significant figures) The probability that the truck comes from Y, given that it arrived late, is 0.349. Send me the next one 📸