**Geometric and statistical probability**

# Geometric probability

Classical probability required that the number of cases be finite. However, this is not the case in many probability events. For example, within one hour, there are an infinite number of different moments of time. In this case, we use a measure and talk about geometric probability. The geometric measure can be, for example, the length of a time interval, the length of a distance, the area, the volume, the magnitude of the angle, etc. The probability of event A is then calculated:

**Example 1**. A bus always departs every fifteen minutes and takes in passengers from the stop every three minutes before departure. What is the probability that a passenger will have to wait more than four minutes to get on the bus when he comes to a stop without knowing the schedule?

Let's draw the situation:

**Answer**: A passenger who occasionally arrives at a stop will have to wait more than four minutes to get on the bus with a probability of 0,53.

**Example 2**

On a traditional cottage dartboard, the central target has a value of ten points, surrounded by rings of other values down to one. The diameter of the ten is 36 mm and the width of each ring is 18 mm. What is the probability that the value of a dart, thrown with closed eyes, when it hits the board is

a) at least 7

b) exactly 7

a) The radius of the whole dartboard is 180 mm and the radius of the circle corresponding to the result “at least 7” is 72 mm, so

Answer: The probability of getting at least seven is 0,16.

b)

**Answer: **The probability of getting exactly seven is 0.07.

**Example 3.**

In windy conditions, the old spruce in the yard of Sven-Ulf Heiskanen's house swayed menacingly in every direction. What is the probability of the tree hitting the house when it falls when the perpendicular distance of the tree from the house is 8 meters, the width of the wall of the house is 20 meters and the length of the tree is 12 meters.

We make an model picture of the situation and verify by calculating that the picture is correctly drawn.

Let's first find out what is the length of the house wall that the tree hits. If the length is less than 20 meters, which is the width of the house, then the image is correctly drawn. Denote the width of the wall of the house by a, then with Pythagorean theorem

So the picture is drawn correctly. Lets solve the size of the favorable central angle 2α by trigonometry.

Now we can find out the probability that the tree will fall towards the house. The probability is obtained by the quotient of the favorable center angle and the center angle of a full circle.

**Answer:** The tree hits the house with a probability of 0,27

**Statistical probability**

**Statistical probability**

Statistical probability is based on statistical material. Statistical material can give the numbers of proportions of a phenomenon, either in numbers or in relative percentages. The different statistical proportions of the whole random phenomenon are called frequency when quantities are expressed in numbers, and as relative frequency when quantities are expressed as relative percentages.

**Example 4.** The table shows the ages of residents in the Ala-Ylä-Härmälä municipality using statistics.

a) What is the probability that a randomly selected resident is 60 years of age or older?

b) What is the probability that a randomly selected resident is a female aged 20-40?

c) How many percent more likely is a randomly selected female over the age of 60 to be over 80 years old than that a randomly selected male over the age of 60 is over 80 years old?

a)

Answer: A randomly selected resident is over 60 with a probability of 0.46

b)

Answer: The probability that a randomly selected resident is between 20-40 years and female is 0,05

c)

Next, we find out how many percent more likely it is that the selected resident is female

Answer: 91% more likely the selected female is over 80 than the selected male to be over 80

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