To find the probability that the first student chosen is a girl and the second is a boy, we can use the multiplication rule of probability. The multiplication rule states that if two events, A and B, are independent, then the probability of both events occurring is the product of the probabilities of each event occurring individually.
Let's denote the following events:
- A: Choosing a girl first.
- B: Choosing a boy second.
Step 1: Calculate the probability of event A (choosing a girl first).
Since there are 5 girls out of 10 students, the probability of choosing a girl first is:
[tex]\[ P(A) = \frac{\text{Number of girls}}{\text{Total number of students}} = \frac{5}{10} = \frac{1}{2} \][/tex]
Step 2: Calculate the probability of event B (choosing a boy second), given that a girl has already been chosen.
After choosing one girl, there are 9 students left (5 are boys and 4 are girls). So the probability of picking a boy next is:
[tex]\[ P(B|A) = \frac{\text{Number of boys remaining}}{\text{Total number of students remaining}} = \frac{5}{9} \][/tex]
Step 3: Multiply the probabilities of A and B to get the combined probability.
Using the rule of multiplication for independent events, the probability of both A and B occurring is:
[tex]\[ P(A \text{ and } B) = P(A) \times P(B|A) = \left(\frac{1}{2}\right) \times \left(\frac{5}{9}\right) \][/tex]
Step 4: Simplify the result, if necessary.
[tex]\[ P(A \text{ and } B) = \frac{1 \times 5}{2 \times 9} = \frac{5}{18} \][/tex]
Therefore, the probability that the first student chosen is a girl and the second is a boy is [tex]\(\frac{5}{18}\)[/tex] in simplest form.
