### Video Transcript

A real number π is randomly chosen between 10 and 100. Find the probability that π is closer to 50 than 100.

A real number is a value of a continuous quantity that can represent a distance along a line. It includes positive or negative numbers, whole numbers, or decimal numbers. As well as all of these rational numbers, it includes all the irrational numbers. We are told that π is a randomly chosen real number between 10 and 100. Letβs consider the number line between these two values.

We need to identify all the numbers that are closer to 50 than 100. Clearly, all the real numbers that lie between 10 and 50 inclusive will be closer to 50. These numbers can be represented by the inequality π is greater than or equal to 10 and less than or equal to 50. The midpoint between 50 and 100 is 75. This means that all those numbers less than 75 will be closer to 50 and all the numbers greater than 75 will be closer to 100. These two can be written as inequalities. When π is greater than 50 and less than 75, it will be closer to 50. And when π is greater than 75 and less than or equal to 100, it will be closer to 100.

As we are dealing with all the real numbers, all numbers up to 74.999 and so on or 74.9 recurring will be less than 75. This means that we need to set aside the infinitesimally small region where π equals 75. As weβre including all the real numbers, the chance of being exactly 75 is incredibly small. We, therefore, need to consider the regions to the left and right of 75. From 10 to 75, we have a distance or difference of 65. And from 75 to 100, we have a difference or distance of 25.

The probability of a random event occurring can be written as a fraction where the numerator is the number of successful outcomes and the denominator is the number of possible outcomes. A successful outcome in this case is when π is closer to 50. This is equal to 65. The number of possible outcomes is the total range from 10 to 100 which is 90. This could also be found by adding 65 and 25. The numbers 65 and 90 are both divisible by five. 65 divided by five is equal to 13, and 90 divided by five is 18. Therefore, the fraction 65 over 90 can be simplified to 13 over 18.

The probability that a randomly chosen number between 10 and 100 is closer to 50 than 100 is 13 out of 18 or thirteen eighteenths.