If is confirm that two thirds of the students of a college are females, calculate the probability that on entering the college, the first two students one meet are
(i) the same gender
(ii) different gender
m - m ... 1/3 * 1/3 = 1/9
m - f ... 1/3 * 2/3 = 2/9
f - m ... 2/3 * 1/3 = 2/9
f - f ... 2/3 * 2/3 = 4/9
(i) 1/9 + 4/9 = 5/9
(ii) 2/9 + 2/9 = 4/9
To calculate the probability of certain events happening, we need to divide the number of favorable outcomes by the total number of possible outcomes.
Let's assume there are a total of n students in the college. Given that two-thirds of the students are females, we can determine the number of females in the college as (2/3)n, and the number of males as (1/3)n.
(i) Probability of meeting two students of the same gender:
For the first student, there are two possibilities: either a female or a male. Let's calculate the probability of meeting a female first:
Probability of meeting a female = (Number of females) / (Total number of students)
= (2/3)n / n
= 2/3
If the first student is a female, then for the second student to be a female as well, the probability would be:
Probability of the second student being female = (Number of females - 1) / (Total number of students - 1)
= (2/3)n - 1 / n - 1
= (2/3)
Now, since the events of the first and second students being female are independent, the probability of both events happening is the product of their individual probabilities:
Probability of both students being female = Probability of first student being female * Probability of second student being female
= (2/3) * (2/3)
= 4/9
(ii) Probability of meeting two students of different genders:
The probability of the first student being female is still 2/3.
If the first student is a female, then the probability of the second student being male is:
Probability of the second student being male = (Number of males) / (Total number of students - 1)
= (1/3)n / (n - 1)
= 1/3
Since the events of the first student being female and the second student being male are independent, the probability of both events happening is the product of their individual probabilities:
Probability of one female and one male student = Probability of first student being female * Probability of second student being male
= (2/3) * (1/3)
= 2/9
Therefore, the probability of meeting the first two students who are of the same gender is 4/9, while the probability of meeting two students of different genders is 2/9.