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.
ScanSolve AI Answer
Step 1: Determine the total number of tubers, spoilt tubers, and good tubers. Total tubers = 15 Spoilt tubers = 5 Good tubers = Total tubers - Spoilt tubers = 15 - 5 = 10 Step 2: Calculate the total number of ways to choose 3 tubers from 15. This is given by the combination formula nk = (n!)/(k!(n-k)!). Total ways to choose 3 tubers = 153 = (15!)/(3!(15-3)!) = (15!)/(3!12!) = (15 × 14 × 13)/(3 × 2 × 1) = 5 × 7 × 13 = 455 a) Find the probability that none is spoilt. Step 3: Calculate the number of ways to choose 3 tubers such that none are spoilt. This means all 3 chosen tubers must be good. We choose 3 good tubers from the 10 good tubers. Ways to choose 3 good tubers = 103 = (10!)/(3!(10-3)!) = (10!)/(3!7!) = (10 × 9 × 8)/(3 × 2 × 1) = 10 × 3 × 4 = 120 Step 4: Calculate the probability that none is spoilt. P(none spoilt) = Ways to choose 3 good tubersTotal ways to choose 3 tubers = (120)/(455) = (24)/(91) The probability that none is spoilt is (24)/(91). b) Find the probability that all are spoilt. Step 5: Calculate the number of ways to choose 3 tubers such that all are spoilt. This means all 3 chosen tubers must be spoilt. We choose 3 spoilt tubers from the 5 spoilt tubers. Ways to choose 3 spoilt tubers = 53 = (5!)/(3!(5-3)!) = (5!)/(3!2!) = (5 × 4)/(2 × 1) = 10 Step 6: Calculate the probability that all are spoilt. P(all spoilt) = Ways to choose 3 spoilt tubersTotal ways to choose 3 tubers = (10)/(455) = (2)/(91) The probability that all are spoilt is (2)/(91). c) Find the probability that exactly one is spoilt. Step 7: Calculate the number of ways to choose 3 tubers such that exactly one is spoilt. This means we choose 1 spoilt tuber from 5 spoilt tubers AND 2 good tubers from 10 good tubers. Ways to choose 1 spoilt tuber = 51 = 5 Ways to choose 2 good tubers = 102 = (10!)/(2!(10-2)!) = (10!)/(2!8!) = (10 × 9)/(2 × 1) = 45 Ways to choose exactly one spoilt = 51 × 102 = 5 × 45 = 225 Step 8: Calculate the probability that exactly one is spoilt. P(exactly one spoilt) = Ways to choose exactly one spoiltTotal ways to choose 3 tubers = (225)/(455) = (45)/(91) The probability that exactly one is spoilt is (45)/(91). d) Find the probability that at least one is spoilt. Step 9: Calculate the probability that at least one is spoilt. The event "at least one is spoilt" is the complement of the event "none is spoilt". P(at least one spoilt) = 1 - P(none spoilt) Using the result from part (a): P(at least one spoilt) = 1 - (24)/(91) = (91 - 24)/(91) = (67)/(91) The probability that at least one is spoilt is (67)/(91). 3 done, 2 left today. You're making progress.