a bag contains eight yellow marbles, nine green marbles, three purple marbles, and five red marbles. three…

Explanation:

Step1: Calculate total number of marbles

$8 + 9+3 + 5=25$ marbles.

Step2: Define the probability formula for combinations

The probability $P(X = k)=\frac{C(n_1,k)\times C(n_2,n - k)}{C(n,n)}$, where $n$ is the total number of marbles, $n_1$ is the number of purple marbles, $n_2$ is the number of non - purple marbles, $n$ is the number of marbles chosen, and $k$ is the number of purple marbles chosen. We want $P(X\leq1)=P(X = 0)+P(X = 1)$.

Step3: Calculate $P(X = 0)$

$n_1 = 3$ (purple marbles), $n_2=22$ (non - purple marbles), $n = 3$. $C(3,0)=\frac{3!}{0!(3 - 0)!}=1$, $C(22,3)=\frac{22!}{3!(22 - 3)!}=\frac{22\times21\times20}{3\times2\times1}=1540$, $C(25,3)=\frac{25!}{3!(25 - 3)!}=\frac{25\times24\times23}{3\times2\times1}=2300$. $P(X = 0)=\frac{C(3,0)\times C(22,3)}{C(25,3)}=\frac{1\times1540}{2300}\approx0.6696$.

Step4: Calculate $P(X = 1)$

$C(3,1)=\frac{3!}{1!(3 - 1)!}=3$, $C(22,2)=\frac{22!}{2!(22 - 2)!}=\frac{22\times21}{2\times1}=231$. $P(X = 1)=\frac{C(3,1)\times C(22,2)}{C(25,3)}=\frac{3\times231}{2300}\approx0.2996$.

Step5: Calculate $P(X\leq1)$

$P(X\leq1)=P(X = 0)+P(X = 1)\approx0.6696 + 0.2996=0.9692\approx0.971$.

Answer:

$0.971$