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. (c) Find the probability that Michael wins all three prizes.

To solve this problem, we will first outline the total number of tickets sold and the number of tickets Michael bought. Then, we will create a tree diagram and calculate the probabilities for each scenario. ### (a) Tree Diagram The tree diagram represents the possible outcomes for Michael winning prizes. Since there are three prizes, we can represent the outcomes as follows: 1. **First Prize (Motorbike)**: Win (W) or Lose (L) 2. **Second Prize (Bicycle)**: Win (W) or Lose (L) 3. **Third Prize (Television)**: Win (W) or Lose (L) The tree diagram would look like this: ``` Start / \ W L / \ W L / \ W L / \ W L / \ W L / \ W L / \ W L / \ W L ``` Each branch represents winning or losing a prize. The total outcomes can be calculated based on the number of tickets Michael has. ### (b) 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 the first prize: \( \frac{250 - 8}{250} = \frac{242}{250} \) - Probability of losing the second prize: \( \frac{250 - 8}{250} = \frac{242}{250} \) - Probability of losing the third prize: \( \frac{250 - 8}{250} = \frac{242}{250} \) Since these events are independent, the probability that Michael wins no prize is: \[ P(\text{No Prize}) = P(L) \times P(L) \times P(L) = \left(\frac{242}{250}\right)^3 \] Calculating this: \[ P(\text{No Prize}) = \left(\frac{242}{250}\right)^3 = \frac{242^3}{250^3} \approx 0.925 \] ### (c) 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 two can be calculated as follows: 1. Choose 1 prize to win (3 ways to choose which prize). 2. Probability of winning that prize: \( \frac{8}{250} \) 3. Probability of losing the other two prizes: \( \left(\frac{242}{250}\right)^2 \) Thus, the probability of winning exactly one prize is: \[ P(\text{One Prize}) = 3 \times \left(\frac{8}{250}\right) \times \left(\frac{242}{250}\right)^2 \] Calculating this: \[ P(\text{One Prize}) = 3 \times \frac{8}{250} \times \left(\frac{242}{250}\right)^2 \] ### (d) 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 2 prizes). 2. Probability of winning those 2 prizes: \( \left(\frac{8}{250}\right)^2 \) 3. Probability of losing the other prize: \( \frac{242}{250} \) Thus, the probability of winning exactly two prizes is: \[ P(\text{Two Prizes}) = 3 \times \left(\frac{8}{250}\right)^2 \times \left(\frac{242}{250}\right) \] ### (e) Probability that Michael wins all three prizes To find the probability that Michael wins all three prizes, we can calculate it as follows: \[ P(\text{All Three Prizes}) = \left(\frac{8}{250}\right)^3 \] ### Summary of Probabilities 1. **No Prize**: \( P(\text{No Prize}) = \left(\frac{242}{250}\right)^3 \) 2. **One Prize**: \( P(\text{One Prize}) = 3 \times \left(\frac{8}{250}\right) \times \left(\frac{242}{250}\right)^2 \) 3. **Two Prizes**: \( P(\text{Two Prizes}) = 3 \times \left(\frac{8}{250}\right)^2 \times \left(\frac{242}{250}\right) \) 4. **All Three Prizes**: \( P(\text{All Three Prizes}) = \left(\frac{8}{250}\right)^3 \) You can plug in the values to get the numerical probabilities.