The numbers 1, 2, 3, 4, 5 and 6 are each written on separate cards - HSC - SSCE Mathematics Standard - Question 17 - 2022 - Paper 1

To complete the tree diagram, we compare each of Amy's cards with Bob's cards as follows:

• When Amy chooses card 1:

  • Bob chooses card 2: Bob wins.
  • Bob chooses card 4: Bob wins.
  • Bob chooses card 6: Bob wins.

• When Amy chooses card 3:

  • Bob chooses card 2: Amy wins.
  • Bob chooses card 4: Bob wins.
  • Bob chooses card 6: Bob wins.

• When Amy chooses card 5:

  • Bob chooses card 2: Amy wins.
  • Bob chooses card 4: Amy wins.
  • Bob chooses card 6: Bob wins.

The completed tree diagram is:

Amy's card    Bob's card
     1         2  B
               4  B
               6  B
     3         2  A
               4  B
               6  B
     5         2  A
               4  A
               6  B

To find the probability that Bob wins:

• Total outcomes = 9 (3 cards from Amy × 3 cards from Bob)
• Winning outcomes for Bob = 6.

Thus, the probability that Bob wins is:

$$P(Bob ext{ wins}) = \frac{6}{9} = \frac{2}{3}$$