A school holds a raffle to raise funds for the matric dance. A total of 250 tickets are sold. There are three prizes to be won: First prize-Motorbike Second prize-Bicycle Third prize-television Michael buys 8 tickets. He could win all three prizes. (a) Draw a tree diagram. (b) Find the probability that Michael wins no prize. (c) Find the probability that Michael wins one prize. (d) Find the probability that Michael wins two prizes. (e) Find the probability that Michael wins all three prizes.

To solve this problem, we will first outline the situation and then address each part of the question step by step. ### Total Tickets and Prizes - Total tickets sold: 250 - Tickets bought by Michael: 8 - Total prizes: 3 (Motorbike, Bicycle, Television) ### (a) Draw a Tree Diagram A tree diagram for this scenario would have three branches for the three prizes. Each branch would further split into two branches: one for winning the prize and one for not winning the prize. However, since I cannot draw here, I will describe it: 1. Start with a single point (Michael's ticket). 2. From this point, draw three branches for each prize: - First Prize (Motorbike) - Win - Lose - Second Prize (Bicycle) - Win - Lose - Third Prize (Television) - Win - Lose Each of these branches represents the outcome of winning or losing each prize. ### (b) Find the Probability that Michael Wins No Prize To find the probability that Michael wins no prize, we first need to calculate the probability of losing each prize. - Probability of losing a prize = (Total tickets - Michael's tickets) / Total tickets - Probability of losing a prize = (250 - 8) / 250 = 242 / 250 = 121 / 125 Since the events are independent, the probability that Michael loses all three prizes is: \[ P(\text{No Prize}) = P(\text{Lose Motorbike}) \times P(\text{Lose Bicycle}) \times P(\text{Lose Television}) = \left(\frac{121}{125}\right)^3 \] Calculating this: \[ P(\text{No Prize}) = \left(\frac{121}{125}\right)^3 = \frac{121^3}{125^3} = \frac{1771561}{1953125} \approx 0.905 \] ### (c) Find the Probability that Michael Wins One Prize To find the probability that Michael wins exactly one prize, we can use the binomial probability formula. The probability of winning one prize and losing the other two can be calculated as follows: 1. Choose 1 prize to win (3 ways to choose which prize). 2. Probability of winning that prize = Michael's tickets / Total tickets = 8 / 250 = 4 / 125. 3. Probability of losing the other two prizes = (242 / 250) for each. Thus, the probability of winning exactly one prize is: \[ P(\text{One Prize}) = \binom{3}{1} \times P(\text{Win}) \times P(\text{Lose})^2 \] \[ = 3 \times \left(\frac{8}{250}\right) \times \left(\frac{242}{250}\right)^2 \] Calculating this: \[ = 3 \times \frac{8}{250} \times \left(\frac{242}{250}\right)^2 = 3 \times \frac{8}{250} \times \frac{58564}{62500} \] \[ = 3 \times \frac{8 \times 58564}{250 \times 62500} = 3 \times \frac{468512}{15625000} \approx 0.090 \] ### (d) Find the Probability that Michael Wins Two Prizes To find the probability that Michael wins exactly two prizes, we can use a similar approach: 1. Choose 2 prizes to win (3 ways to choose which prizes). 2. Probability of winning those prizes = (8 / 250) for each. 3. Probability of losing the other prize = (242 / 250). Thus, the probability of winning exactly two prizes is: \[ P(\text{Two Prizes}) = \binom{3}{2} \times P(\text{Win})^2 \times P(\text{Lose}) \] \[ = 3 \times \left(\frac{8}{250}\right)^2 \times \left(\frac{242}{250}\right) \] Calculating this: \[ = 3 \times \left(\frac{8}{250}\right)^2 \times \left(\frac{242}{250}\right) = 3 \times \frac{64}{62500} \times \frac{242}{250} \] \[ = 3 \times \frac{64 \times 242}{62500 \times 250} = 3 \times \frac{15488}{15625000} \approx 0.003 \] ### (e) Find the Probability that Michael Wins All Three Prizes To find the probability that Michael wins all three prizes, we can calculate it directly: \[ P(\text{All Three Prizes}) = P(\text{Win Motorbike}) \times P(\text{Win Bicycle}) \times P(\text{Win Television}) \] \[ = \left(\frac{8}{250}\right)^3 = \frac{512}{15625000} \approx 0.0000328 \] ### Summary of Probabilities - Probability of winning no prize: \( \approx 0.905 \) - Probability of winning one prize: \( \approx 0.090 \) - Probability of winning two prizes: \( \approx 0.003 \) - Probability of winning all three prizes: \( \approx 0.0000328 \) These calculations provide a comprehensive overview of Michael's chances in the raffle.