Before getting into the concept Probability let’s know about Chance
Chance:
- There are scenarios in our life that are certain to happen, while some that are impossible to happen and some that may or may not happen. The situation that may or may not happen has a chance of happening.
For example:
1. You’re older today than yesterday (Certain to happen)
2. Tomorrow will be a cloudy day (Can happen but not certain)
3. A dice when tossed shall land up with 8 on top (Impossible)
Five terms related to Probability you should know
- Trial: A trial is an action that results in one or several outcomes.
For example: When you throw two coins the possible outcomes are TT, HH, HT, and TH.
- Experiment: An experiment is an investigation or a procedure that has its varying results. The number of possible results are called outcomes.
- Random experiments and outcomes: An experiment in which the result of the trail cannot be predicted in advance.
For example: To understand and measure the chance we can perform the experiments like tossing a coin, rolling a dice and spinning the spinner etc.
When we toss a coin we have only two possible results, head or tail. The chance of getting either is the same and you cannot say what you would get. Such tossing is like an experiment known as a “random experiment”.
The outcomes of random experiments may be equally likely or may not be. In the coin-tossing experiment head or tail are two possible outcomes.
- Event: An event associated with the random experiments is the collection of some outcomes of the experiment.
For example: Throwing a die, getting a prime number is an event consisting of 2, 3, and 5.
- Equally likely outcomes: The outcomes of an experiment are equally likely if each has the same chance of occurring.
For example: When we toss a coin or roll a dice, we assume that the coin and the die are fair and unbiased i.e., for each toss or roll the chance of all possibilities is equal.
- Sample space: The set of all possible outcomes of one or more events is called the sample space. You can use tables or tree diagrams to find the sample space of two or more events.
For example: Sample space for choosing a card from the deck of cards.
What is Probability?
Definition: The chance of an event occurring is called the probability of the event happening. It tells us how likely it is for an event to happen; it does not tell us what is going to happen. There is an even chance of an event happening (happen/not happen).
- MOST LIKELY, I will watch the cricket match.
- There is NO CHANCE for me to watch the cricket match.
- Both teams have EQUAL CHANCE to win the toss.
In many situations we make such statements and use our past experience and logic to make decisions. However, the decisions may not always favor us. Thus we make a decision by guessing the future happening, that is whether an event occurs or not. Our decisions can either work or not. We try to measure numerically the chance of occurrence or non-occurrence of some events just as we measure many other things in our daily life. This kind of measurement helps us make decisions in a more systematic manner. Therefore we study probability to figure out the chance of something happening. Before measuring numerically the chance of happening we grade it using the following terms given in the table.
Let’s observe the table below:
Getting head/tail
- Probabilities can be written as decimals or fractions. As the number of trials increases, the probability of all equally likely outcomes comes very close to each other.
- The probability of an event A
P (A) = \(\frac{\text{Number of favorable outcomes}}{\text{Number of total possible outcomes}}\)
- Probabilities are between 0 and 1, including 0 and 1. They can be written as fractions, decimals, or percents.
- The probability of an event that is certain = 1
- The probability of an event that is impossible = 0
- The probability of an event always lies between 0 and 1 (0 and 1 inclusive)
- Relative frequency: It is the measure of probability. When you conduct an experiment, the relative frequency of an event is the fraction or percent of the time that the event occurs.
Relative frequency = \(\frac{\text{Number of times the event occurs}}{\text{Total number of times you conduct the Experiment}}\)
Uses of Probability in real life
- The meteorological department predicts the weather by observing trends from the data collected over many years in the past.
- Insurance companies calculate the probability of an accident or casualty to determine insurance premiums.
- “An exit poll” is taken after the election. This involves asking voted people to which party they have voted. This gives an idea of the winning chances of each candidate and predictions are made accordingly.
Solved Probability Examples
Example 1: You flip a bottle and record the number of times it lands upright and the number of times it lands on its side. Describe the likelihood that the bottle lands upright on your next flip.
Solution:
The bottle landed upright 5 times in a total of 30 flips.
Relative frequency \(~=~\frac{5}{30}\) (the bottle landed upright 5 times / there was a total of 30 flips)
The relative frequency is \(~=~\frac{5}{30}~=~\frac{1}{6}\) or 16%, so it is unlikely that the bottle lands upright.
Example 2: If two identical coins are tossed for one time simultaneously. Find the possible outcomes, the number of total outcomes.
Solution: (a) The possible outcomes are
Coin 1 | Coin 2 |
|---|---|
Head | Head |
Head | Tail |
Tail | Head |
Tail | Tail |
(b) Number of possible outcomes is 4
As we know,
Probability of getting two heads = Number of favorable outcomes of getting two heads / number of possible outcomes \(~=~\frac{1}{4}\)
Probability of getting at least one head 1\(~=~\frac{3}{4}\)
Probability of getting no heads \(~=~\frac{1}{4}\)
Probability of getting only one head \(~=~\frac{2}{4}~=~\frac{1}{2}\)
Example 3: A student brought a very large jar of animal crackers to share with students in class. Rather than count and sort all the different types of crackers, the student randomly chose 20 crackers and found the following counts for the different types of animal crackers. Estimate the probability of selecting a zebra.
Animal | Number selected |
|---|---|
Lion | 2 |
Camel | 1 |
Monkey | 4 |
Elephant | 5 |
Zebra | 3 |
Penguin | 3 |
Tortoise | 2 |
Total | 20 |
Solution: The estimated probability of picking a zebra is \(~=~\left (\frac{3}{20} \right )\) or 0.15 or 15%
- This means that an estimate of the proportion of the time a zebra will be selected is 0.15 or 15% of the time. This could be written as P(zebra) = 0.15, or the probability of selecting a zebra is 0.15.
Example 4: A student randomly selected crayons from a large bag of crayons. The table below shows the number of each color crayon in the bag. Now, suppose the students were to randomly select one crayon from the bag.
Color | Number |
|---|---|
Brown | 10 |
Blue | 5 |
Yellow | 3 |
Green | 3 |
Orange | 3 |
Red | 6 |
Solution: Probability of Selecting a blue crayon \(= \frac{5}{30}= \frac{1}{6}=0.167\) or \(16.7\%\)
Probability of Selecting a brown crayon \(= \frac{10}{30}= \frac{1}{3}=0.333 \) or \(33.3\%\)
Probability of Selecting a red crayon or yellow crayon \(= \frac{9}{30}= \frac{3}{10}=0.3 \) or \(30\%\)
Probability of selecting a pink crayon \(= \frac{0}{30}= 0\% \)
We try to measure numerically the chance of occurrence or non-occurrence of some events just as we measure many other things in our daily life. Therefore we study probability to figure out the chance of something happening. This kind of measurement helps us to make decisions in a more systematic manner.
There are scenarios in our life that are certain to happen, while some that are impossible to happen and some that may or may not happen. The situation that may or may not happen has a chance of happening. And this chance of happening is represented in numerical form which indicates the probability.
It is the measure of probability. When you conduct an experiment, the relative frequency of an event is the fraction or percent of the time that the event occurs.
Probability can be applied regularly in everyday life like to toss a coin during cricket matches, to predict the weather conditions by meteorological departments, to determine the insurance premiums by insurance companies and during the elections to predict the winner probability can be applied.
The terms used to represent the chance of happening is certain, more likely, equally likely, less likely, impossible. By using these terms the probability is estimated accordingly.
No, the probability of an event occurring always lies between 0 to 1. It should not be a negative number. And also the sum of probabilities is always equal to one.
Probability of an event A is
P(A) = \(\frac{\text{Number of favorable outcomes}}{\text{Number of total possible outcomes}}\)