# Question Video: Determining the Total Number of Marbles in a Bag Using Probabilities Mathematics

A bag contains an unknown number of marbles. There are 2 red marbles, some white marbles, and some black marbles. The probability of getting a white marble is 1/3, and the probability of getting a black marble is 1/2. Calculate the number of marbles in the bag.

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### Video Transcript

A bag contains an unknown number of marbles. There are two red marbles, some white marbles, and some black marbles. The probability of getting a white marble is one-third, and the probability of getting a black marble is one-half. Calculate the number of marbles in the bag.

We are told that a bag contains three different colored marbles. They are red, white, and black. We are told that there are two red marbles. However, we are not told in the question the number of white or black marbles. We are told though that the probability of selecting a white marble is one-third and the probability of selecting a black marble is one-half. Our aim in this question is to calculate the total number of marbles in the bag. We recall that the probability of any event occurring lies between zero and one inclusive. We also know that the sum of the probabilities of all outcomes of an event must equal one.

This means that the probability of selecting a red marble plus the probability of selecting a white marble plus the probability of selecting a black marble must equal one. We know that the probability of selecting a white marble is one-third and the probability of selecting a black marble is one-half. We know that one-third is equivalent to two-sixths and one-half is equivalent to three-sixths. This means that one-third plus one-half is equal to five-sixths. The probability of selecting a red marble plus five-sixths is equal to one. We can then subtract five-sixths from both sides of this equation such that the probability of selecting a red marble is one-sixth.

As the probability of an event can be written as the number of favorable outcomes over the total number of outcomes and we know that there are two red marbles in the bag, we need to find a fraction equivalent to one-sixth with a numerator of two. The fraction one-sixth is equal to two over what. As we have multiplied the numerator by two, we need to do the same to the denominator. One-sixth is equal to two twelfths. This suggests that the total number of marbles in the bag is 12.

We can check this by now working out the number of white and black marbles. One-third of 12 is equal to four. Therefore, there are four white marbles. One-half of 12 is equal to six, so there are six black marbles in the bag. As two plus four plus six equals 12, we can therefore conclude that the number of marbles in the bag is 12. It is also worth noting that the fractions two twelfths, four twelfths, and six twelfths are equivalent to one-sixth, one-third, and one-half, respectively.