Let's break down the step-by-step solution to determine the probability that if three students are chosen at random from the math class, the first student chosen to be in the group is in the fourth period science class, but the other two are not.
We need to calculate the probability for the following sequence:
1. The first student chosen is in the science class.
2. The second student chosen is not in the science class.
3. The third student chosen is not in the science class.
### Step-by-Step Calculation
1. Probability that the first student is in the science class:
- There are 8 students out of 30 in the science class.
- The probability of selecting a science student first is [tex]\( \frac{8}{30} \)[/tex].
2. Probability that the second student is not in the science class:
- After selecting the first student, there are 29 students remaining.
- Out of the 29 students, 30 - 8 = 22 students are not in the science class.
- The probability of selecting a non-science student second is [tex]\( \frac{22}{29} \)[/tex].
3. Probability that the third student is not in the science class:
- After selecting the first and second students, there are 28 students remaining.
- Out of these 28 students, 22 - 1 = 21 students are still not in the science class.
- The probability of selecting a non-science student third is [tex]\( \frac{21}{28} \)[/tex].
### Total Probability
To find the total probability of this specific sequence of events happening, we multiply the probabilities of each individual event:
[tex]\[ \text{Total Probability} = \left(\frac{8}{30}\right) \times \left(\frac{22}{29}\right) \times \left(\frac{21}{28}\right) \][/tex]
### Conclusion
Given that the options are:
- [tex]\(\left(\frac{8}{30}\right)\left(\frac{22}{29}\right)\left(\frac{21}{28}\right)\)[/tex]
- [tex]\(\left(\frac{8}{30}\right)\left(\frac{8}{30}\right)\left(\frac{8}{30}\right)\)[/tex]
- [tex]\(\left(\frac{8}{30}\right)\left(\frac{7}{29}\right)\left(\frac{6}{28}\right)\)[/tex]
- [tex]\(\left(\frac{8}{30}\right)\left(\frac{22}{30}\right)\left(\frac{21}{30}\right)\)[/tex]
The correct expression for the probability is:
[tex]\[ \left(\frac{8}{30}\right)\left(\frac{22}{29}\right)\left(\frac{21}{28}\right) \][/tex]
This matches our computed probabilities, confirming that the correct answer is:
[tex]\[ \left(\frac{8}{30}\right)\left(\frac{22}{29}\right)\left(\frac{21}{28}\right) \][/tex]
The final numerical answer for the total probability is approximately 0.1517.
