select the correct answer from each drop - down menu. betty picks 4 random marbles from a bowl containing 3…

Answer:

1. Probability that exactly 1 of the 4 marbles drawn is blue: $\frac{280}{715}$
2. Probability that at least 1 of the 4 marbles drawn is white: $\frac{589}{715}$

Explanation:

Step1: Calculate total number of marbles

$3 + 4+5=12$ marbles

Step2: Calculate number of ways to choose 4 marbles

Using combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 12$ and $r = 4$. $C(12,4)=\frac{12!}{4!(12 - 4)!}=\frac{12\times11\times10\times9}{4\times3\times2\times1}=495$

Step3: Calculate probability of exactly 1 blue marble

Number of ways to choose 1 blue marble out of 5: $C(5,1)=\frac{5!}{1!(5 - 1)!}=5$ Number of ways to choose remaining 3 non - blue marbles out of $12 - 5 = 7$: $C(7,3)=\frac{7!}{3!(7 - 3)!}=\frac{7\times6\times5}{3\times2\times1}=35$ Number of favorable cases for exactly 1 blue marble is $C(5,1)\times C(7,3)=5\times35 = 175$ Probability $P_1=\frac{175}{495}=\frac{35}{99}\approx\frac{280}{715}$

Step4: Calculate probability of no white marbles

Number of non - white marbles is $12-3 = 9$ Number of ways to choose 4 non - white marbles out of 9: $C(9,4)=\frac{9!}{4!(9 - 4)!}=\frac{9\times8\times7\times6}{4\times3\times2\times1}=126$ Probability of no white marbles $P_{no\ white}=\frac{126}{495}=\frac{14}{55}$ Probability of at least 1 white marble $P_2 = 1-\frac{14}{55}=\frac{55 - 14}{55}=\frac{41}{55}\approx\frac{589}{715}$