We have 20 pupils in class, 12 girls and 8 boys. We arrange the pupils in a row, and now need to calculate the following probability:
a. The probability that Jana, one of the girls, will not stand next to another girl.
b. 4 pupils out of the 20 (12 girls and 8 boys) are randomly chosen for the class commitee, What is the probability that both genders will have representation in the commitee.
I tried many times with no success.
Thanks in advance!
Best Answer
For (a) there are two cases: Jana is at one end, or she isn't.
If she's at the end, pick which end ($2$), then pick which of the eight boys stands next to her ($8$). Then place the other $18$ children ($18!$).
If she's not at the end, pick where she stands ($18$), then pick the boy that stands on her left ($8$) and then on her right ($7$). Then place the other $17$ children ($17!$).
Hence the probability is
$$P_a = \frac{2 \cdot 8 \cdot 18! + 18 \cdot 8 \cdot 7 \cdot 17!}{20!} = \frac{18}{95}.$$
For (b) there are three cases:
So in order:
Hence:
$$P_b = \frac{8 (_{12}C_3) + 12 (_{8}C_3) + (_{12}C_2) (_{8}C_2)}{_{20}C_4}.$$