Question

There are $10$ red marbles and $8$ green marbles in a jar. if you take three marbles from the jar (without replacement), the probability that they are all red is___.

Answer 1

Find the probability that all marbles are red:

Number of red marbles is $10$.

Number of green marbles is $8$.

Total number of marbles$=10+8=18$.

We know that,

$\mathrm{Probability}=\frac{\mathrm{Favorable}\mathrm{outcomes}}{\mathrm{Total}\mathrm{outcomes}}$.

First $1$ out of $10$ red marbles is taken. Then the probability is $P\left(R_1\right)=\frac{10}{18}$.

If one ball is taken out then, the number of red balls are $\left(10-1\right)=9$.

The total number of marbles left are $\left(18-1\right)=17$.

Then the probability is $P\left(R_2\right)=\frac{9}{17}$.

If one more ball is taken out then, the number of red balls are $\left(9-1\right)=8$.

The total number of marbles left are $\left(17-1\right)=16$.

Then the probability is $P\left(R_3\right)=\frac{8}{16}$.

Since all the three marble are drawn same time, all events are happening at the same time. Hence, by the multiplication rule of probability, the probability of all marbles being red is,

$\begin{array}{rcl}P\left(R\right)& =& \frac{10}{18}\times \frac{9}{17}\times \frac{8}{16}\\ \therefore P\left(R\right)& =& 0.147\end{array}$

Hence, the probability that all marbles are red is $0.147$.