Question Video: Using a Venn Diagram to Determine the Probability of the Complement of an Event Mathematics

The days in a certain month are classified as rainy days, hot days, both rainy and hot days, or neither rainy nor hot days. Let 𝑅 denote rainy days and let 𝐻 denote hot ones. Use the Venn diagram below to calculate the probability that a day is not rainy.

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Video Transcript

The days in a certain month are classified as rainy days, hot days, both rainy and hot days, or neither rainy nor hot days. Let 𝑅 denote rainy days and let 𝐻 denote hot ones. Use the Venn diagram below to calculate the probability that a day is not rainy.

The Venn diagram shown has two circles representing rainy and hot days. The number in the intersection of both circles represents days that were both rainy and hot. And the number outside the two circles represents the number of days that were neither rainy nor hot. In this question, we are asked to calculate the probability that a day is not rainy.

Let’s begin by calculating the total number of days in the month. This is the sum of nine, four, two, and 15. As this is equal to 30, there are 30 days in the given month. The circle that represents rainy contains a four and a two. There were four days that were just rainy and two days that were rainy and hot. This gives a total of six rainy days. The probability that a day is rainy is therefore equal to six out of 30. Both the numerator and denominator here are divisible by six. This means that the fraction simplifies to one-fifth. And the probability that a day is rainy is one-fifth.

We could work out the number of days that are not rainy by adding those numbers outside of the circle in the Venn diagram. Alternatively, we can use our knowledge of the complement of an event denoted 𝐴 bar or 𝐴 prime. The probability of 𝐴 bar is equal to one minus the probability of 𝐴. This means that the probability that a day is not rainy is one minus one-fifth, which is equal to four-fifths. This fraction is equivalent to 24 over 30, which is the number of days it did not rain over the total number of days.

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