Handling Data 2
5 marks · HardA bag contains 4 white balls and 5 black balls. Two balls are drawn one after another without replacement.
Draw a tree diagram to show all the possible outcomes of selecting each ball in order of being drawn (white/black).
Calculate the probability that both balls are the same colour.
What is the probability that at least one white ball is drawn?
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) — · must show First branch: white (4/9), black (5/9); second branches: if white then W again (3/8) / B (5/8); if black then B again (4/8), W (4/8)
(b) 37/72
(c) 52/72 or 13/18
Must show: Branches: first pick white/black, then conditional probabilities for second
Common mistakes: Treating as replacement; Order error in probabilities
Show mark scheme and worked solution
Marking points
D1 (a) · 2 marks · Correct two-stage tree showing correct branches and labelled probabilities.
M1 (b) · 1 mark · Calculates probability WW and BB and adds them; uses correct fractions.
A1 (b) · 1 mark · Gets final correct probability value for both same colour. (follow-through)
A1 (c) · 1 mark · Correctly finds probability for at least one white: (1 - P(both black)). (follow-through)
Worked solution
a) Tree diagram structure: - First pick: White (W, 4/9), Black (B, 5/9) - Second pick: If W: W (3/8), B (5/8) If B: B (4/8), W (4/8)
b) Probability both same: WW: (4/9)*(3/8) = 12/72 BB: (5/9)*(4/8) = 20/72 Total = 12/72 + 20/72 = 32/72 = 8/18 = 4/9
(Updated after checking, total probability both same = 12/72 (WW) + 20/72 (BB) = 32/72 = 8/18 ~ 0.444)
c) At least one white: 1 - P(both black) = 1 - 20/72 = 52/72 = 13/18 ~ 0.722