There are 5 red marbles and 4 blue marbles in a bag. Bill and Ben are playing a game in which they take turns drawing a marble from the bag and then replacing it.

To win the game, n must draw a red marble and for Bill to win he must draw a blue marble. They continue taking turns until there is a winner. Ben goes first

Find the probability that Ben wins on his first draw.

Find the probability that Ben wins in three or less of his turns.

Find the probability that Ben wins the game.

Alice and Raoul take turns throwing darts at a dartboard. The winner is whoever hits the bullseye first. Alice has a $\frac{1}{30}$ chance of hitting the bullseye, while Raoul has a $\frac{1}{40}$
40 chance of hitting the bullseye. Alice throws the first dart.
(a) Draw a tree diagram for the first four throws of the games (two throws for Alice and two for Raoul) 

(b) What is the probability that Alice wins on her first or second throw? 
(c) What is the probability that Alice will eventually win the game. 

 

$$\begin{gathered} Harry lives in Homebush and is starting a new job in the city. He needs to catch a train to \\ get to work. His new boss says that he cannot be late on the first two days of his new job \\ or he will lose it. The probability that his train will arrive on time is 0.96 \\ What is the probability that Harry’s train is late on the first day? \\ What is the probability of the train being late on the first two days? \\ What is the probability of Harry keeping his job? \\ What is the probability that Harry arrives late on exactly one of the first \\ three days of his new job? (do not round off your answer). \end{gathered}$$

$$\begin{gathered} A game is played in which two coloured dice are thrown once. The six faces of the red die are \\ numbered 3, 5, 7, 8, 9 and 11. The six faces of the white die are numbered 1, 2, 4, 6, 10 and 12. \\ The player wins if the number on the white die is larger than the number on the red die. What is \\ the probability that the player wins once in two successive games? \end{gathered}$$

$$\begin{gathered} Dani and Chris used the spinner shown above to play a game. Dani spun the spinner twice and added the results of the two spins to get his score. Chris then took his turn. The player with the highest score won the game. \\ Use a tree diagram or a sample space to show all the possible scores Dani could have achieved when he played the game. \\ What is the probability that Dani scored 6 in the game? \\ Dani’s score was 6. What is the probability that Chris won the game? \end{gathered}$$

$$\begin{gathered} On the compass diadram below, Mary is at position A, 25km due north of positionB. John is at \\ B. Mary walks towards B at 4km/h. John moves due west at 6km/h. \\ \left(i\right) Show that the dis\tan ce between Mary and John after t hours is given by: \\ {d}^2=52t^2-200t+625 \\ \left(ii\right) Letting L=d^2,find the time when L is minmum. \\ \left(iii\right) Hence find the minimum dis\tan ce between John and Mary, correct to the nearest \\ kilometre. \end{gathered}$$

$$\begin{gathered} The rate of increase of a population P\left(t\right) of people in a certain city is \\ governed by the equation \frac{dP}{dt}=kP where k is a cons\tan t and t is the \\ time in years. The population of the city doubles every twenty years. \\ \left(i\right) Show that k=\frac{1}{20}\ln 2. \\ \left(ii\right) In which year will the city reach a population three times that \\ which it had at the beginning of 2006? \\ \left(iii\right) If at the beginning of 2010 the population is 20 million, what \\ will be the population at the beginning of year 2060? \end{gathered}$$

$$\begin{gathered} Given the sequence In 4, \ln 16, \ln 64...... \\ \left(i\right) Show that it is arithmetic. \\ \left(ii\right) Hence, find the sum of the first 20 terms in exact form. \end{gathered}$$

$$\begin{gathered} Chairs are arranged in rows in front of a stage in a concert hall, so the row closest to the stage is the \\ first row. Each row has two more chairs than the row in front of it. There are forty-two chairs in the \\ tenth row. \\ \left(i\right) How many chairs are in the first row? \\ \left(ii\right) The seating arrangement has a total of 680 chairs. \\ How many rows of chairs are in the concert hall? \\ \left(iii\right) How many chairs are in the last row? \end{gathered}$$

$Find the number of the terms in an arthmetic sequence with a=5, d=2 and the last term 43.$