We are given 10 red and 15 green balls. we are asked to draw two balls successively. What is the probability that one ball is red and other is green.

Question

My effort for this question was

Since I have to select two balls so my sample space will be = $25C2$ and the favourable case is: $10C1$ $15C1$ because I have $10$ red balls and $15$ green balls, so my probability is $\dfrac{10C1\space 15C1}{25C2}$.

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but the answer is something like this : since we have to select two balls so sample space would be: $25 \cdot 24$ and the favourable case is : $10C1\space 15C1 + 15C1\space 10C1$.

so the probability would become : $\dfrac{10C1 \space 15C1 + 15C1 \space 10C1}{25 \cdot 24}$.

Can anybody please guide why the answer is like this?

* Thanks in Advance *

Answer 1

There are $10\cdot 15$ combinations of one red ball and one green ball. There are a total of $25$ choose $2$ pairs to pick from, and $150$ them are "successful". $25$ choose $2$ is $300$. Thus the probability of success is $\frac{150}{300}=\frac{1}{2}$.

Answer 2

Your answer is correct and is equivalent to the given answer.

If we select the balls sequentially, the first ball can be selected in $25$ ways, which leaves $24$ ways to select the second ball since the first ball is not replaced. There are two favorable cases:

1. A red ball is selected first, then a green ball is selected second: There are $10$ ways to select the red ball and $15$ ways to select the green ball, so there are $10 \cdot 15$ such cases.
2. A green ball is selected first, then a red ball is selected second: There are $15$ ways to select the green ball and $10$ ways to select the red ball, so there are $15 \cdot 10$ such cases.

In total, the number of favorable cases is $10 \cdot 15 + 15 \cdot 10$. Thus, the probability that one red ball and one green ball are selected when two balls are selected successively from $10$ red and $15$ green balls is $$\frac{10 \cdot 15 + 15 \cdot 10}{25 \cdot 24} = \frac{2 \cdot 10 \cdot 15}{25 \cdot 24} = \frac{10 \cdot 15}{\frac{25 \cdot 24}{2}} = \frac{\binom{10}{1}\binom{15}{1}}{\binom{25}{2}}$$ The author(s) took order of selection into account, while you did not.