Math Help - Probability

1. Probability

1.Out of 5 mathematics teachers and 7 English teachers a committee consisting of 2 mathematics teachers and 3 English teachers is to be formed. In how many ways can this be done if any of the mathematics teachers and any English teachers can be chosen ?

2.A bag contains 5 white balls, 3 black balls and 2 green balls. A ball is chosen at random from the bag and not replaced. in three draws find the probability of obtaining white, black and green in that order.

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2. Re: Probability

For 1: You have 5 math teachers, and you want to choose two of them. Using binomial coefficient notation, there are $\binom{5}{2}$ ways to choose the two math teachers. Next, you want three English teachers. The choice of English teacher is independent from the choice of math teacher. So, you can apply the product principle. There are $\binom{7}{3}$ ways to choose the three English teachers. The total number of committees is given by the product of those two.

2. For the first ball, your chances of drawing a white ball are $\dfrac{5}{5+3+2} = \dfrac{1}{2}$. Now there is one less ball in the mix. Your chances of drawing a black ball are $\dfrac{3}{4+3+2} = \dfrac{1}{3}$. Finally, the probability that the final ball is green is $\dfrac{2}{4+2+2} = \dfrac{1}{4}$. So, by the product principle, the probability is the product of those three probabilities.