Question Video: Determining Conditional Probability Using a Two-Way Table | Nagwa Question Video: Determining Conditional Probability Using a Two-Way Table | Nagwa

Question Video: Determining Conditional Probability Using a Two-Way Table Mathematics • Third Year of Secondary School

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A children’s entertainer has two boxes. Box 𝐴 contains 1 red balloon, 2 green balloons, and 7 black balloons. Box 𝐵 contains 4 red balloons, 4 green balloons, and 2 black balloons. By making a two-way table, calculate the probability that the selected balloon is not red if it is drawn from box 𝐵.

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Video Transcript

A children’s entertainer has two boxes. Box 𝐴 contains one red balloon, two green balloons, and seven black balloons. Box 𝐵 contains four red balloons, four green balloons, and two black balloons. By making a two-way table, calculate the probability that the selected balloon is not red if it is drawn from box 𝐵.

We’ve been told to approach this problem by making a two-way table. This is a way of sorting data based on two characteristics. Let’s think about the characteristics of the data that have been given here.

The children’s entertainer has two boxes: box 𝐴 and box 𝐵. So the first characteristic is which box the balloon is in. In each box there are red, green, and black balloons. So the second characteristic is the color of the balloon. We can therefore set up a two-way table in which the columns represent the boxes and the rows represent the color of the balloons.

We then complete the table using the information in the question. Box 𝐴 contains one red balloon. Box 𝐴 also contains two green balloons, and seven black balloons. In box 𝐵, there are four red balloons, four green balloons, and two black balloons.

It’s useful to also add the row and column totals, as well as the overall total. Summing down the columns, we find that there are 10 balloons in box 𝐴 and 10 balloons in box 𝐵. And then summing across the rows, we find that there are five red balloons in total, six green balloons in total, and nine black balloons in total. The overall total can be found by summing either the row or column totals, and in both cases, it gives 20.

Now that we’ve completed the two-way table, we’re ready to calculate the required probability. The probability we’re asked for is the probability that the selected balloon is not red if it is drawn from box 𝐵. This means that we’re not actually interested in the balloons in box 𝐴 at all. So instead of the probability being out of the total of 20 balloons, it will actually just be out of the 10 balloons in box 𝐵. Of the 10 balloons in box 𝐵, a total of six of them are not red, so the probability will be six-tenths.

We can denote this probability using the notation for a conditional probability, which is a vertical line. We read this as the probability that the balloon is not red, given that it was taken from box 𝐵.

Finally, we need to give this fraction in its simplest terms, so we divide both the numerator and denominator by two. By making a two-way table, we’ve calculated that the probability that the selected balloon is not red if it is drawn from box 𝐵 is three-fifths.

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