A mathematics club is made up of 10 boys and 15girls .The club has 3 officials.by the help of a tree diagram,find the probability that:(a)the club officials are all girls (b) at least one of the officials is a boy
You can visualize this problem with a tree diagram, but I will explain it here without drawing the diagram.
There are a total of 25 students in the club, and three positions to fill for club officials. We can calculate the probabilities step by step according to the different scenarios.
(a) Probability that all club officials are girls:
For the first official, the probability of choosing a girl is 15/25. After choosing one girl, there are 14 girls and 10 boys left, and 24 students in total. So for the second official, the probability of choosing a girl is 14/24. After choosing the second girl, there are 13 girls and 10 boys left, and 23 students in total. For the third official, the probability of choosing a girl is 13/23.
Therefore, the probability of choosing all three officials as girls is (15/25)*(14/24)*(13/23) = 2730/13800 which simplifies to 91/460.
(b) At least one official is a boy:
To find the probability of at least one official being a boy, we can first calculate the probability of all officials being girls (which we found in part (a) to be 91/460) and subtract this from 1.
Probability of at least one official being a boy = 1 - Probability of all officials being girls = 1 - 91/460 = 369/460.
So, the probability of having at least one boy in the club officials is 369/460.