Question

Two identical bags A and B contained identical balls. Bag A has 5 yellow balls and 2 red balls. Bag B has 4 yellow balls and 3 red balls. Two balls are drawn from each bag without replacement.

1. Draw a tree diagram to illustrate the above information  
2. Find the probability that;
   1. Both balls are yellow  
   2. The first ball is red and the second yellow   
   3. A red and yellow balls are picked  
   4. At least one ball is red  

Answer 1

1.  
   
2.   
   1. Both balls are yellow
      P(AYY) or P(BYY)
      (½ × ⁵/₇ × ⁴/₆) = (½ × ⁴/₇ × ³/₆)
      ⁵/₁₂ + ¹/₇ = ⁸/₂₁
   2. First ball is red, second ball yellow
      P(ARY) or P(BRY)
      (½ × ²/₇ × ⁵/₆) + (½ × ³/₇ × ⁴/₆)
      ⁵/₄₂ + ¹/₇ = ¹¹/₄₂
   3. P(AYR) or P(ARY) or P(BYR) or P(BRY)
      (½ × ⁵/₇ × ²/₆) + (½ × ²/₇ × ⁵/₆) + (½ × ⁴/₇ × ³/₆) + (½ × ³/₇ × ⁴/₆)
      ⁵/₄₂ + ⁵/₄₂ + ¹/₇ + ¹/₇ = ¹¹/₂₁
   4. At least one ball is red
      P(AYR) or P(ARY) or P(BYR) or P(BRY) or P(ARR) or P(BRR)
      (½ × ⁵/₇ × ²/₆) + (½ × ²/₇ × ⁵/₆) + (½ × ⁴/₇ × ³/₆) + (½ × ³/₇ × ⁴/₆) + (½ × ²/₇ × ¹/₆)+(½ × ³/₇ × ²/₆)
      ⁵/₄₂ + ⁵/₄₂ + ¹/₇ + ¹/₇ + ¹/₄₂ + ¹/₁₄ =¹³/₂₁