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 — a) i. All three balls have the same colour. Step 1: Calculate the total number of ways to draw 3 balls from the bag. There are 5 blue + 4 green + 3 yellow = 12 balls in total. The number of ways to choose 3 balls from 12 is given by the combination formula nk = (n!)/(k!(n-k)!). Total ways = 123 = (12 × 11 × 10)/(3 × 2 × 1) = 220 Step 2: Calculate the number of ways to draw three balls of the same colour. This can happen in three mutually exclusive ways: all three are blue, or all three are green, or all three are yellow. • Number of ways to draw 3 blue balls: 53 = (5 × 4 × 3)/(3 × 2 × 1) = 10 • Number of ways to draw 3 green balls: 43 = (4 × 3 × 2)/(3 × 2 × 1) = 4 • Number of ways to draw 3 yellow balls: 33 = (3 × 2 × 1)/(3 × 2 × 1) = 1 The total number of ways for all three balls to have the same colour is the sum of these possibilities: Favorable ways = 10 + 4 + 1 = 15 Step 3: Calculate the probability. P(all three same colour) = Favorable waysTotal ways = (15)/(220) = (3)/(44) The probability that all three balls have the same colour is (3)/(44). a) ii. Two balls have the same colour. Step 1: The total number of ways to draw 3 balls remains the same as in part (a) i. Total ways = 123 = 220 Step 2: Calculate the number of ways to draw exactly two balls of one colour and one ball of a different colour. This involves choosing 2 balls of one colour and 1 ball from the remaining colours. • 2 Blue and 1 Green: 52 × 41 = ((5 × 4)/(2 × 1)) × 4 = 10 × 4 = 40 • 2 Blue and 1 Yellow: 52 × 31 = ((5 × 4)/(2 × 1)) × 3 = 10 × 3 = 30 • 2 Green and 1 Blue: 42 × 51 = ((4 × 3)/(2 × 1)) × 5 = 6 × 5 = 30 • 2 Green and 1 Yellow: 42 × 31 = ((4 × 3)/(2 × 1)) × 3 = 6 × 3 = 18 • 2 Yellow and 1 Blue: 32 × 51 = ((3 × 2)/(2 × 1)) × 5 = 3 × 5 = 15 • 2 Yellow and 1 Green: 32 × 41 = ((3 × 2)/(2 × 1)) × 4 = 3 × 4 = 12 The total number of ways for exactly two balls to have the same colour is the sum of these possibilities: Favorable ways = 40 + 30 + 30 + 18 + 15 + 12 = 145 Step 3: Calculate the probability. P(two balls same colour) = Favorable waysTotal ways = (145)/(220) = (29)/(44) The probability that two balls have the same colour is (29)/(44). What's next?