Answer :
We have a jar with 25 marbles: 5 yellow, 4 orange, 2 red, 6 black and 8 blue.
We have to calculate the probability of the following event: "Draw a red marble, then a yellow one and then a black one" without replacement.
The events are independent so we can calculate the probability of this events as the product of three events (one per draw).
Each event probability can be estimated as the number of marbles of the color we want to draw divided by the number of marbles left.
Then, the first event, "draw a red marble" will have a probability of 2/25.
The second event, "draw a yellow marble", will have a probability 5/24. The denominator is now 24 as we have one (red) marble less in the jar to pick from.
The third and last event, "draw a black marble", will have a probability of 6/23.
Then, we can calculate the probability of all this three events in a row as:
[tex]\begin{gathered} P(x_1=r,x_2=y,x_3=b)=P(x_1=r)\cdot P(x_2=y)\cdot P(x_3=b) \\ P(x_1=r,x_2=y,x_3=b)=\frac{2}{25}\cdot\frac{5}{24}\cdot\frac{6}{23} \\ P(x_1=r,x_2=y,x_3=b)=\frac{60}{13800} \\ P(x_1=r,x_2=y,x_3=b)\approx0.00435 \end{gathered}[/tex]Answer: the probability is approximately 0.00435.
