A group of 5 boys and 4 girls take part in a study on the type of plants found in a reserved forest area. Each day, two pupils are chosen at random to write a report.
a) Calculate the probability that both pupils chosen to write the report on the first day are boys.
b) Two boys wrote the report on the first day. They are then exempted from writing the report on the second day.
Calculate the probability that both pupils chosen to write the report on the second day are of the same gender.
To solve both parts of this problem, we'll need to use concepts from probability and combinations.
a) To calculate the probability that both pupils chosen to write the report on the first day are boys, we need to consider the total number of possibilities and the number of favorable outcomes.
There are a total of 9 pupils (5 boys + 4 girls) in the group. Since two pupils are chosen at random, the total number of possible outcomes is given by selecting 2 out of the 9 pupils, which can be calculated using combinations.
The number of ways to choose 2 boys out of the 5 available is given by the combination formula C(5, 2) = 10.
So, the total number of possible outcomes is C(9, 2) = 36 (choosing 2 out of the 9 pupils).
The number of favorable outcomes, where both pupils chosen are boys, is 10.
Therefore, the probability that both pupils chosen to write the report on the first day are boys is 10/36, which simplifies to 5/18.
b) After two boys have written the report on the first day, there are now 3 boys and 4 girls remaining in the group. Since we again need to select 2 pupils at random, we can calculate the probability that both pupils chosen on the second day are of the same gender.
The total number of possible outcomes is C(7, 2) = 21 (choosing 2 out of the 7 remaining pupils).
The number of favorable outcomes, where both pupils chosen are boys or both are girls, is given by the number of ways to choose 2 boys out of the 3 available (C(3, 2) = 3) plus the number of ways to choose 2 girls out of the 4 available (C(4, 2) = 6). So, the number of favorable outcomes is 3 + 6 = 9.
Therefore, the probability that both pupils chosen to write the report on the second day are of the same gender is 9/21, which simplifies to 3/7.