To find the median of a discrete probability distribution, we need to look at the cumulative probabilities and determine at which point the cumulative probability reaches or exceeds 0.5.
Given the distribution of scores and their corresponding probabilities:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline \text{Score} & 1 & 2 & 3 & 4 & 5 \\
\hline \text{Probability} & 0.18 & 0.20 & 0.26 & 0.21 & 0.15 \\
\hline
\end{array}
\][/tex]
First, we will calculate the cumulative probabilities:
1. Cumulative probability for score 1:
[tex]\[ \text{P}(X \leq 1) = 0.18 \][/tex]
2. Cumulative probability for score 2:
[tex]\[ \text{P}(X \leq 2) = 0.18 + 0.20 = 0.38 \][/tex]
3. Cumulative probability for score 3:
[tex]\[ \text{P}(X \leq 3) = 0.38 + 0.26 = 0.64 \][/tex]
4. Cumulative probability for score 4:
[tex]\[ \text{P}(X \leq 4) = 0.64 + 0.21 = 0.85 \][/tex]
5. Cumulative probability for score 5:
[tex]\[ \text{P}(X \leq 5) = 0.85 + 0.15 = 1.00 \][/tex]
Now, let's summarize the cumulative probabilities in a table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline \text{Score} & 1 & 2 & 3 & 4 & 5 \\
\hline \text{Cumulative Probability} & 0.18 & 0.38 & 0.64 & 0.85 & 1.00 \\
\hline
\end{array}
\][/tex]
Next, we need to identify the score at which the cumulative probability reaches or exceeds 0.5. Looking at the cumulative probabilities:
- For score 1: 0.18 (less than 0.5)
- For score 2: 0.38 (less than 0.5)
- For score 3: 0.64 (greater than 0.5)
The cumulative probability reaches 0.64 at score 3, which is the first time it exceeds 0.5.
Therefore, the median of the distribution is:
[tex]\[ \boxed{3} \][/tex]
