Question

formula 2 points
a box contains 7 orange pencils, 4 yellow pencils, and 3 green pencils. two pencils are selected, one at a time, with replacement. find the probability that the first pencil is green and the second pencil is yellow.
express your answer as a decimal, rounded to the nearest hundredth.
formula 2 points
a gumball machine contains 12 red gumballs and 3 white gumballs. two gumballs are purchased, one after the other, without replacement. find the probability that at least one gumball is white.

Explanation:

Step1: Calculate total number of pencils

Total pencils = \(7 + 4+3=14\)

Step2: Calculate probability of first pencil being green

Probability of first green \(P(G_1)=\frac{3}{14}\)

Step3: Calculate probability of first not green and second yellow

Probability of first not green \(P(\overline{G_1})=\frac{11}{14}\), probability of second yellow \(P(Y_2)=\frac{4}{14}\)
Since with replacement, these are independent events. Probability \(P(\overline{G_1}\cap Y_2)=\frac{11}{14}\times\frac{4}{14}=\frac{44}{196}\)

Step4: Calculate required probability

\(P = P(G_1)+P(\overline{G_1}\cap Y_2)=\frac{3}{14}+\frac{44}{196}=\frac{42 + 44}{196}=\frac{86}{196}\approx0.44\)

Answer:

\(0.44\)