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4. It has been established that the probabilities of Exater High School (EHS) and Kabi Day School (KDS) soccer teams scoring 0, 1, 2, or 3 goals against each other are as shown below:

[tex]\[
\begin{tabular}{|c|c|c|}
\hline
\textbf{No. of Goals} & \textbf{Probabilities of EHS scoring} & \textbf{Probabilities of KDS scoring} \\
\hline
0 & $3.0 \times 10^{-1}$ & $2.0 \times 10^{-1}$ \\
\hline
1 & $3.0 \times 10^{-1}$ & $4.0 \times 10^{-1}$ \\
\hline
2 & $3.0 \times 10^{-1}$ & $3.0 \times 10^{-1}$ \\
\hline
3 & $1.0 \times 10^{-1}$ & $1.0 \times 10^{-1}$ \\
\hline
\end{tabular}
\][/tex]

(a) Using a tree diagram or otherwise, show all the possible outcomes.
(3 Marks)



Answer :

GinnyAnswer
To determine the possible outcomes of the number of goals scored by Exater High School (EHS) and Kabi Day School (KDS), and their respective probabilities, we need to consider all combinations of scores both teams can achieve, as well as the combined probabilities of these outcomes.

Here are the steps:

### Step 1: Identify the Goal Probabilities
From the given data, the probabilities are:

| No. of Goals | EHS Probability | KDS Probability |
| ------------ | --------------- | --------------- |
| 0 | 0.3 | 0.2 |
| 1 | 0.3 | 0.4 |
| 2 | 0.3 | 0.3 |
| 3 | 0.1 | 0.1 |

### Step 2: List All Possible Outcomes
Each team's goals range from 0 to 3. The outcomes represent the pairings of EHS and KDS goals.

The possible outcomes are:
1. EHS scores 0 goals, KDS scores 0 goals: (0, 0)
2. EHS scores 0 goals, KDS scores 1 goal: (0, 1)
3. EHS scores 0 goals, KDS scores 2 goals: (0, 2)
4. EHS scores 0 goals, KDS scores 3 goals: (0, 3)
5. EHS scores 1 goal, KDS scores 0 goals: (1, 0)
6. EHS scores 1 goal, KDS scores 1 goal: (1, 1)
7. EHS scores 1 goal, KDS scores 2 goals: (1, 2)
8. EHS scores 1 goal, KDS scores 3 goals: (1, 3)
9. EHS scores 2 goals, KDS scores 0 goals: (2, 0)
10. EHS scores 2 goals, KDS scores 1 goal: (2, 1)
11. EHS scores 2 goals, KDS scores 2 goals: (2, 2)
12. EHS scores 2 goals, KDS scores 3 goals: (2, 3)
13. EHS scores 3 goals, KDS scores 0 goals: (3, 0)
14. EHS scores 3 goals, KDS scores 1 goal: (3, 1)
15. EHS scores 3 goals, KDS scores 2 goals: (3, 2)
16. EHS scores 3 goals, KDS scores 3 goals: (3, 3)

### Step 3: Calculate the Probabilities of Each Outcome
To calculate the probability of each outcome, we multiply the probability of EHS's goals by the probability of KDS's goals.

The calculated probabilities are:
1. [tex]\((0, 0)\)[/tex]: [tex]\(0.3 \times 0.2 = 0.06\)[/tex]
2. [tex]\((0, 1)\)[/tex]: [tex]\(0.3 \times 0.4 = 0.12\)[/tex]
3. [tex]\((0, 2)\)[/tex]: [tex]\(0.3 \times 0.3 = 0.09\)[/tex]
4. [tex]\((0, 3)\)[/tex]: [tex]\(0.3 \times 0.1 = 0.03\)[/tex]
5. [tex]\((1, 0)\)[/tex]: [tex]\(0.3 \times 0.2 = 0.06\)[/tex]
6. [tex]\((1, 1)\)[/tex]: [tex]\(0.3 \times 0.4 = 0.12\)[/tex]
7. [tex]\((1, 2)\)[/tex]: [tex]\(0.3 \times 0.3 = 0.09\)[/tex]
8. [tex]\((1, 3)\)[/tex]: [tex]\(0.3 \times 0.1 = 0.03\)[/tex]
9. [tex]\((2, 0)\)[/tex]: [tex]\(0.3 \times 0.2 = 0.06\)[/tex]
10. [tex]\((2, 1)\)[/tex]: [tex]\(0.3 \times 0.4 = 0.12\)[/tex]
11. [tex]\((2, 2)\)[/tex]: [tex]\(0.3 \times 0.3 = 0.09\)[/tex]
12. [tex]\((2, 3)\)[/tex]: [tex]\(0.3 \times 0.1 = 0.03\)[/tex]
13. [tex]\((3, 0)\)[/tex]: [tex]\(0.1 \times 0.2 = 0.02\)[/tex]
14. [tex]\((3, 1)\)[/tex]: [tex]\(0.1 \times 0.4 = 0.04\)[/tex]
15. [tex]\((3, 2)\)[/tex]: [tex]\(0.1 \times 0.3 = 0.03\)[/tex]
16. [tex]\((3, 3)\)[/tex]: [tex]\(0.1 \times 0.1 = 0.01\)[/tex]

### Final Outcome Listing
Combining the outcomes with their probabilities, we get:

[tex]\[ \begin{align*} \text{Outcomes: } & [(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 1), (2, 2), (2, 3), (3, 0), (3, 1), (3, 2), (3, 3)] \\ \text{Probabilities: } & [0.06, 0.12, 0.09, 0.03, 0.06, 0.12, 0.09, 0.03, 0.06, 0.12, 0.09, 0.03, 0.02, 0.04, 0.03, 0.01] \end{align*} \][/tex]

This comprehensive list details all possible outcomes of goals scored by both teams, along with the associated probabilities for each outcome.

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