Video: Solving Problems Using Theoretical Probability

A bag contains white, red, and black balls. The probability of drawing a white ball at random is 11/20 and a red ball at random is 3/10. What is the smallest number of red balls and black balls that could be in the bag?

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Video Transcript

A bag contains white, red, and black balls. The probability of drawing a white ball at random is 11 out of 20 and a red ball at random is three out of 10. What is the smallest number of red balls and black balls that could be in the bag?

We are told in the question that there are three different color balls in the bag. The probability of selecting a white ball is 11 out of 20 or eleven twentieths. The probability of selecting a red ball is three out of 10 or three-tenths. We are not given the probability of selecting a black ball.

In order to compare fractions, we need to ensure that the denominators are the same. The lowest common multiple of 10 and 20 is 20, so we need to multiply the denominator of the second fraction by two. Whatever we do to the denominator, we must do to the numerator. Three multiplied by two is equal to six, and 10 multiplied by two is 20. Therefore, the fractions three-tenths and six twentieths are equivalent.

We know that the sum of all probabilities is one. In this question, as our denominator is 20, this is equal to twenty twentieths. Eleven twentieths plus six twentieths is equal to seventeen twentieths. When the denominators are the same, we just add the numerators. Subtracting this from one or twenty twentieths gives us three twentieths. This means that the probability of selecting a black ball is three twentieths.

We now have three probabilities that we can compare as the denominators are all the same. The ratio of white to red to black balls is 11 to 6 to three. This means that the smallest number of balls in total is 20, where 11 would be white, six red, and three black. The smallest number of red balls and black balls that could be in the bag are six and three, respectively.

As we’re only given the probabilities, the total number of balls could be any multiple of 20. For example, we could have 40 balls in total where 22 are white, 12 are red, and six are black. However, as we were looking for the smallest number of red and black balls, the correct answer is six red and three black.

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