If you flip three fair coins, what is the probability that you’ll get all three heads?

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Probability is one of the important chapters in maths and is quite interesting. It has a lot of real-life implementations. But some students find it difficult because of a lack of clarity of concepts. They get confused while solving various types of problems from this chapter. To avoid such issues, students need the guidance of teachers.

Often, in schools, the maths class duration is insufficient for an in-depth study of the chapters. Children rely on coaching institutes. Since the pandemic broke out, the coaching classes are being conducted online to ensure safety. Cuemath is one such platform that helps students to learn even the most difficult chapters in maths easily. In this article, we will focus on all aspects of the probability chapter in maths.

Is probability difficult to understand?

How can I avoid confusion while solving probability problems?

Are there any tricks for solving questions from this chapter easily?

These are just a few questions, and since we know that many more such questions would be popping up in your mind, especially in the minds of those students who are somewhat weak in maths, we will clarify them all in the following sections.

Some basics about probability:

In a certain event, the probability of an instance in it is otherwise known as the ‘chance’ of its occurrence. In your day to day life, you must have noticed that any instance has some probability of happening or not happening. When you want to state this in quantitative terms, it is referred to as the probability of the event’s occurrence.

Terms you need to know before learning probability:

i. Experiment:

It is a task that you perform, which leads to some results.

ii. Outcome and its types:

The result of an experiment is termed as its outcome. Among these outcomes, you are interested in a specific outcome, referred to as a “favorable outcome” among the set of all possible probabilities.

Let us consider an example:

Suppose you have a pack of cards. There are 52 cards in it. If you pick a card and want to know the probability of occurrence of the red king, then:

Number of favorable outcomes= 2

Total possible outcomes = 52

iii. Combination:

The concept of combination comes whenever you are solving the probability question. It means collection. In the case of probability questions, we need to choose something from a set. Hence there has to be clarity in the concept of combination before diving into the probability chapter in maths.

The formula of the number of possible combination:

n C r = n! ÷ [ r! x (n-r)! ]

= [ n x (n-1) x (n-2) x (n-3) x…..x1 ] ÷ [ 1 x 2 x 3 x…..r ] x [(n-r) x ……3 x 2 x 1]

This reduces to:

n C r = [n x (n-1) x (n-2) x (n-3) x……(n-r+1)] ÷ (1 x 2 x 3 x …..x r)

where n indicates the total number of objects present in the set and r is the number of objects you are selecting from the set at a time.

It can also be written as n C r = n C (n-r)

Example:

There are 5 different colored balls in a bag, so the number of ways in which 2 balls can be chosen from it is represented as

5 C 2 = (5 x 4) / (1 x 2) = 10

Solving various types of probability problems:

Here is a list of problem types that you will come across in the probability chapter of; maths and ways of solving them:

i. Using multiplication function:

You have the probability of two or more events. When both these events happen simultaneously, their probability is the multiplication of the given probabilities.

Example:

When A and B both have interviews today, and the chances of A and B clearing it is 1/4 and 1/2 respectively, then the probability of both clearing it can be found out as below:

(1/4) x (1/2) =1/8

ii.Addition of probabilities:

It is also known as the OR function. When we have probabilities of 2 or more events and want to know the chances of either of them happening, can be done by adding the probabilities as below:

Let us consider the example considered in point (i). Here the probability of A or B clearing the interview is:: (1/4) + (1/2) = 3/4

iii. In the example considered in point (i), the probability of A clearing the interview and B not clearing it can be found out as below:

(1/4) x [1 – (1/2)]= 1/8

iv. For finding the probability of non-occurrence of an event, the probability of its occurrence is subtracted from 1. This makes it clear that the probabilities of occurrence and non-occurrence of any event sum up to one.

v. Coin tossing problem:

There are two outcomes of tossing a coin, i.e. head or tail, so the number of events is two. Let us see some examples of problems related to this.

· Probability of getting a tail on tossing one coin is 1/2.

Let us consider a case when 3 fair coins are tossed. Let us consider the head outcomes as H and tail outcomes as T.

So all the possible outcomes = 8 which can be written as= (HHH), (HHT,), (HTH), (THH), (HTT), (THT), (TTH), (TTT)

So the probability of getting all three heads, in this case, is 1/8 as there is just 1 case where all the three coins show Head.

Similarly, the probability of getting a minimum of 2 heads is 4/8, which is 1/2.

vi. Card problem:

The combination concept is used in this case. Suppose you are asked to pick a card from the pack of 52 cards, then:

The total number of possible outcomes= 52 when you choose just one card.

Let us see some other examples in this case:

· Probability of occurrence of 2 queens when 2 cards are randomly drawn from the pack :

Favorable outcomes= 2 (out of 4 Queens)

Total possible outcomes=2 (out of the total 52 cards)

So, required probability is (4 C 2 ) / ( 52 C 2)) = 1/221

vii. Dice problem:

In these problems, you are not required to implement the combination concept. Just find out favorable and total possible outcomes. The number of throws or the number of dices determines the possible outcomes

For example:

When the number of throws = 2

and number of dice = 2

Number of possible outcomes = 6 x 6 = 36

Tips for scoring well in probability questions in maths:

Some simple tips that every student needs to remember to excel in maths, especially the probability chapter, are stated below:

i. Try to understand all the fundamentals of the subject thoroughly. For this, you can take the help of your class teachers and teachers in the coaching institute. Never keep the doubts to yourself, otherwise they will build up and form huge hurdles for you in the future.

ii. Practice different types of problems from various books and online sites to ensure complete coverage of other problems.

iii. Permutations and combinations are closely linked with this chapter. You can often expect questions that ask probability and several permutations or combinations, so it is better to understand well how to solve questions belonging to each of these sections.

It often happens that students get confused between these three topics. A clarity in basics can prevent this.

iv. Attend as many mock tests as possible, to be aware of your actual preparation level. Once you come across any issue, it is better to get it immediately clarified by your tutor so that the doubts do not pile up at the end.

Final words:

The probability chapter in maths has numerous real-life implementations no matter which field a student opts for. So, it is essential to be thorough in its basics and practice all types Of Data with Examples to feel confident while appearing in the final exams and easily use it in real life whenever the need arises.

No matter how skillful a teacher is, he should have the potential to explain a topic in the best way to ensure that students understand it easily. All they need to do is choose the right tutor, such as Cuemath, which has ideal teachers who take care of all the needs of students, teach the concepts with clarity to make sure that they can solve any problem from the chapter.