Mathematics is one of the most important subjects to clear either your school exam or any entrance exams that you will appear for later in life. Mathematics topics are not only important to pass your exams but also help solve real-life situations. Read on to find about Geometric Probability.
Many students fear Mathematics; this is mainly because they do not understand the basic concepts, and hence, are unable to solve the problems. But once you understand the basics, it becomes quite easy for you to solve any problems thrown at you. Mathematics becomes quite fun.
In this article, we are going to discuss one of the most important topics from your Mathematics syllabus, which is Geometric Probability, with its theorems and examples to give you a better idea about the topic. If you are struggling to understand the subject, we can help solve all your doubts. So, let’s get started.
- Understanding Probability
In your everyday life, you may come across various situations such as
- Will it rain today?
- I doubt I will clear today’s test.
- Most probably James will win the match.
- The chances of petrol prices hiking again are very high.
- It is a 50-50 chance that India will win the toss in today’s match.
In such situations, words such as “probably”, “doubt”,” most probably”, “chances”, etc. that are used involve a factor of uncertainty, e.g. in a situation of “probably rain”, whether it will rain or not is predicted on similar conditions of the past. This is applicable for other conditions as well. This “doubt” or “possibility” can be measured by Probability in various cases and situations. Probability is extensively used in areas like Physical Sciences, Medical Sciences, Commerce, Weather Forecasting, etc.
- What do You Understand by Probability?
Probability is the possibility of getting the desired outcome from all the probable outcomes. Probability can also be measured as a percentage (%), where 0 implies an impossible outcome and 100% implies a certain outcome.
- If a possibility can never occur, then it’s probability is 0.
- If something might or might not happen, then it’s probability lies between 0 and 1.
- If chances are something will definitely happen, then its probability is 1.
- Probability is a quantitative measure of certainties.
- Experiment: A job which produces outcomes.
- Trial: Performing an event.
Probability = The number of ways of achieving success/ The total number of possible outcomes
Example: The probability of flipping a coin and the possibility of getting heads is ½, as there is only one way of getting heads and the total number of possible outcomes is 2, which is heads and tails. So, P(heads) = ½.
Equation 1: There are 6 balls in a hat. 3 balls are red, 2 balls are yellow, and 1 ball is blue. What are the chances/ probability of picking up a yellow ball?
Ans: The Probability for this question will be the number of yellow balls in the bag divided by the total number of balls in the bag. So, the answer would be 2/6 = ⅓.
Equation 2: Multiple Events: Independent and Dependent events
Here, we will consider a probability of two events happening such as throwing 2 dice on a table at once and considering a probability that both are 6’s. Two outcomes are considered as Independent, if one of the events doesn’t affect the outcome of the other. If we throw two dice, chances of getting a 6 on the second dice are the same as what we get on the first dice, i.e., ⅙.
Whereas in a box with 2 black and 2 red balls, if you have to pick two balls, the probability for the second ball to be a red ball depends on the first ball you pick. If the first ball is red, there would be 1 red and 2 black balls in the bag when you pick the second ball. So, the chances for getting a red ball are ⅓, and if the first ball is black then, there will be 1 black and 2 red balls left. So, the probability for the second ball to be red is 2/4. When an event depends on the outcome of another event, they are called Dependent.
- Geometrical Probability
Geometric Probability is the calculation of the chances that one will hit a particular area of a figure. It is calculated by the desired area divided by the total area. In the case of Geometric Probability, there are always infinite outcomes. In Geometric Probability, points on a segment or in a region of a plane represent outcomes.
P = Measure of a specified part of the region/ Measure of the whole region
Note: Measure means length, area, volume, etc.
Example: If AD is 20, AB = 8, and CD = 7, what is the probability that a point chosen at random on the segment AD will be located on segment BC?
Ans: Here,
P = length/ total length
P = 5/20 = ¼
- Areas of Sectors
- Sector: A region of a circle bounded by a central angle and an intercepted arc.
Area of Sector A = N/ 360 pie r2
- Segment: A region of a circle bounded by an arc or a chord.
- How to Calculate Geometric Probability?
A geometric distribution is based on a binomial process, which is a sequence of independent trials having two probable outcomes. The geometric distribution is used to find probabilities of a specific number of trials that will occur prior to the first successful event. Similarly, you can also use geometric distribution to find out the Probability of the number of failed attempts, which will occur before the first successful event.
To calculate the probability that a particular number of the trial will occur until the first successful result is:
P(X=x) = p(1-p)x-1p for x = 1, 2, 3, …
Here, x is a whole number. There is no maximum specified value for X. It is a geometric random variable and is the number of trials needed until the first successful event where p is the probability of success on a single trial.
- Binomial Distribution Vs Geometric Probability
Binomial Setting: Each observation falls into one of two categories. The probability of success is the same for each observation. All observations are independent. There are a fixed number of observations.
Geometric Setting: Each observation falls into one of two categories. The probability of success is the same for each observation. The observations are all independent. The variable of interest is the number of trials required to obtain the first success.
- Using Geometric Probability in Real-Life Situations
Example 1:
In a game of dart, the diameter of the bull’s eye is 4 cm. The radius of the middle circle (blue) is 6 cm, and the outer circle (pink) is 9 cm. What is the probability that the dart thrown at the board will land anywhere inside the middle blue circle and not hit the bull’s eye?
Ans: The radius of the blue circle is 6 cm
A = (3.14)(6cm)^2
A = 113.04 cm^2
Bull’s eye (the diameter is 4 cm, so it means the radius is 2 cm)
A= (3.14)(2 cm)^2
A= 12.56 cm^2
Now, we have to subtract the area of the bull’s eye from the area of the blue circle to find the probability of the dart landing in the blue circle and not on the bull’s eye.
So, blue circle – bull’s eye
= 113. 04 cm^2 – 12.56 cm^2
= 100. 48 cm^2
The number of favourable outcomes is equal to 100.48 cm^2
Example 2:
You are visiting San Francisco and are taking a trolley ride to a store on the market street. You are supposed to meet your friend at the market store at 3:00 PM. The trolley runs every 10 minutes, and the trip to the store is 8 minutes. You arrive at the trolley stop at 2:48 PM. What is the probability that you will arrive at the store by 3:00 PM?
Ans: To begin, find the maximum amount of time that you can afford to wait for the trolley and still get to the store by 3:00 PM. Because the ride takes 8 minutes and you need to catch the trolley in no less than 8 minutes.
So, you can afford to wait for 4 minutes (2:52 – 2:48 = 4 min).
You can use a line segment to model a probability that the trolley will come within 4 minutes.
We hope that we were able to help you understand the basic concepts of Geometric Probabilities. For the students preparing for upcoming examinations, having sound knowledge of this concept will allow you to score well in the subject. Practising Geometric Probabilities daily with their formulas can help you understand the topic better. You can also solve previous years’ papers/ sample papers to get an idea on the pattern of the questions with marks distribution. These questions on Geometric Probability will seem very easy because of the data simplified with Cuemath; you will retain the concept for longer. Get all the latest questions, answers, and comprehensives notes on their official site.
