On a store counter are exactly two boxes containing only purple marble

Hi rlitagmatstudy,

I would suggest you approach it this way:

The question asks us to select a purple marble from either of the boxes. Here we need to select a box first(either A or B) and then select the marble from the box. So these events are AND events i.e. both need to be done to select a purple marble from the box. For an AND event we multiply the probability of the events. Hence the probability equation can be written as:

P(Selecting a purple marble) = P(selecting a box) * P(selecting purple marble from the selected box)

The probability of selecting either of the box is the same = \(\frac{1}{2}\)

\(= \frac{1}{2} * \frac{30}{50} + \frac{1}{2} * \frac{15}{50} = \frac{9}{20}.\)

Doing it this way will help you take care of the difference in probability of selecting the box as well as difference in probability of selecting the marble. You go the right answer by dividing the total purple marbles by total number of marbles as there were same number of marbles in both the boxes and the probability of selecting either of the box was same.

Another example

Box A - 20 Purple and 20 yellow marbles
Box B - 15 purple and 35 yellow marbles

Using your method will give us the P(selecting a purple marble) = \(\frac{35}{90} = \frac{7}{18}\).

However the right answer would be P(selecting a purple marble) = \(\frac{1}{2} * \frac{20}{40} + \frac{1}{2} * \frac{15}{50}= \frac{2}{5}.\)

Another variation of this question can be where the probability of selecting the boxes may not be the same. Hence you can't take the ratio of total purple marbles to the total marbles to arrive at the probability of selecting a purple marble from both the boxes.

Hope this helps

Regards
Harsh