Question
Ava and Barry play a game with a bag containing one green marble and two red marbles. Each player in turn randomly selects a marble from the bag, notes its colour and replaces it. Ava wins the game if she selects a green marble. Barry wins the game if he selects a red marble. Ava starts the game.
a.Find the probability that Ava wins on her first turn.[1]
b.Find the probability that Barry wins on his first turn.[2]
c.Find the probability that Ava wins in one of her first three turns.[4]
d.Find the probability that Ava eventually wins.[4]
▶️Answer/Explanation
Markscheme
\({\text{P(Ava wins on her first turn)}} = \frac{1}{3}\) A1
[1 mark]
\({\text{P(Barry wins on his first turn)}} = {\left( {\frac{2}{3}} \right)^2}\) (M1)
\( = \frac{4}{9}\;\;\;( = 0.444)\) A1
[2 marks]
\(P\)(Ava wins in one of her first three turns)
\( = \frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3}\) M1A1A1
Note: Award M1 for adding probabilities, award A1 for a correct second term and award A1 for a correct third term.
Accept a correctly labelled tree diagram, awarding marks as above.
\( = \frac{{103}}{{243}}\;\;\;( = 0.424)\) A1
[4 marks]
\({\text{P(Ava eventually wins)}} = \frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3} + \ldots \) (A1)
using \({S_\infty } = \frac{a}{{1 – r}}\) with \(a = \frac{1}{3}\) and \(r = \frac{2}{9}\) (M1)(A1)
Note: Award (M1) for using \({S_\infty } = \frac{a}{{1 – r}}\) and award (A1) for \(a = \frac{1}{3}\) and \(r = \frac{2}{9}\).
\( = \frac{3}{7}\;\;\;( = 0.429)\) A1
[4 marks]
Total [11 marks]