Oliver has 30 marbles, 12 are red, 10 are green and 8 are black. he asks three of his friends to take out a marble and replace it. what is the probability that his friends each take out a different colored marble?

Question

Savannah Anderson · Accepted Answer

To be checked

Let the probability of a color be designated as #P("color")#

Let red be R #->P(R)=12/30#

Let green be G #->P(G)=10/30#
Let black be B #->P(B)=8/30#

These probabilities do not change as you progress through the selection as what is selected is returned to the sample.

#cancel("Each person selects 3 and returns after each selection. ")#

Each person selects 1 and returns it ready for the next person to make their selection.

#color(brown)("All possible success type selection:")#

Note that this diagram is only for the 'success' part. To include the fail part would make the diagram rather large.

So the probability is:

#6xx[8/30xx10/30xx12/30] = 16/75#

Savannah Anderson · Answer

16/75 or 21.3%

This can be divided into two parts. Firstly, what is the likelihood that three balls of different colors will be selected? This is straightforward because the ball is changed every time; the odds of selecting a red ball are 12/30, a blue ball is 10/30, and a black ball is 8/30. Consequently, the probability of selecting three different colored balls is the product of each probability; the sequence is irrelevant. This comes out to be (12/30)x(10/30)x(8/30). The number of ways to choose three different colored balls is now determined by multiplying the number of ways to choose by two, or 3x2x1 = 6. This is because there are three ways to choose the first ball, which can be either red, green, or black; however, there are only two ways to choose the second (since we have already chosen one color, leaving two colors remaining, since each ball must be a different color); and there is only one way to choose the last, which can be determined by the same method. Thus, the total probability equals six times the likelihood of selecting three distinct colored balls (6x(12/30)x(10/30)x(8/30)), which equals the previously mentioned number.

Genesis Allen · Answer

undefined To find the probability that Oliver's friends each take out a different colored marble, we first calculate the probability for each friend selecting a marble of a different color.

The probability that the first friend selects a marble of a different color than the others is calculated as:

$ P(\text{different color for friend 1}) = \frac{{30 - 0}}{{30}} \times \frac{{30 - 12}}{{30}} \times \frac{{30 - 22}}{{30}} = \frac{{18}}{{30}} \times \frac{{18}}{{30}} \times \frac{{8}}{{30}} = \frac{{2592}}{{27000}} $

Similarly, the probability for the second friend and the third friend selecting a different colored marble is:

$ P(\text{different color for friend 2}) = \frac{{18}}{{30}} \times \frac{{18}}{{30}} \times \frac{{8}}{{30}} = \frac{{2592}}{{27000}} $

$ P(\text{different color for friend 3}) = \frac{{18}}{{30}} \times \frac{{18}}{{30}} \times \frac{{8}}{{30}} = \frac{{2592}}{{27000}} $

Therefore, the overall probability that each friend selects a different colored marble is the product of these probabilities:

$ P(\text{each friend selects different colored marble}) = \frac{{2592}}{{27000}} \times \frac{{2592}}{{27000}} \times \frac{{2592}}{{27000}} = \frac{{91136448}}{{729000000}} \approx 0.1252 $