Show work! Suppose a hat contains ten marbles of three different colors. Four ar

Question

Show work!

Suppose a hat contains ten marbles of three different colors. Four are red, five are black, and one is green. Assume all selections are made with replacement (the marble selected is placed back in the hat).

Question 1:    What is the probability that three successive marbles selected will be black?

Question 2:    What is the probability that three successive marbles selected will be green?

Question 3:    What is the probability that for five successive marbles selected, at least one will be green?

Question 4:    What is the probability that three successive marbles selected will each be a different color?

Question 5:    What is the probability that ten successive marbles selected will be black?

Explanation / Answer

Since the selections ae made with replacement the probability will remain same irrespective of order of selection. The probability of selecting a red marble in a single selection would be,

P(red) = 4/10 = 0.4

Similarly the probability of selecting a black/green marbles would be given by,

P(Black) = 5/10 = 0.5

P(green) = 1/10 = 0.1

Question 1:    What is the probability that three successive marbles selected will be black?

Solution :

We will use R,B, and G to represent a Red marble, a black marble, and a green marble respectively. The probability of sekection of each marble is independent of another selection.

The required probability would be given by,

P(BBB) = P(B)*P(B)*P(B) = 0.5*0.5*0.5 = 0.125   

Question 2:    What is the probability that three successive marbles selected will be green?

Solution :

The required probability would be given by,

P(GGG) = P(G)*P(G)*P(G) = 0.1*0.1*0.1 = 0.001

Question 3:    What is the probability that for five successive marbles selected, at least one will be green?

Solution :

The complementary event of selecting atleast one green marble is to select no green marble. The probability of not selecting a green marble in a single selection would be given by,

P (Gc) = 1 - P(G) = 1 - 0.1 = 0.9

The probability of not selecting a green marble while selecting 5 marbles would be given by,

P(GcGcGcGcGc) = P(Gc)*P(Gc)*P(Gc)*P(Gc)*P(Gc) = 0.9*0.9*0.9*0.9*0.9 = 0.59049

The probability of selecting atleast one green marble while selecting 5 marbles would be given by,

P = 1 - 0.59049 = 0.40951

Question 4:    What is the probability that three successive marbles selected will each be a different color?

Solution : There can be 6 different ways when three different marbles are selected. These are RBG, RGB, BRG, BGR, GRB, GBR.

The probability of each way of selection is equal and will be given by,

P(RBG) = 0.4*0.5*0.1 = 0.020

Hence the probability that the three successive marbles selected will be of different color would be given by,

P = 0.020*6 = 0.12

Question 5:    What is the probability that ten successive marbles selected will be black?

Solution :

The probability that ten successive marbles selected will be black would be given by,

P = 0.510 = 0.00098

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