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Probability

Probability of an Event

Let us conduct an experiment. Take a coin and toss it 10 times. See how many times it lands on heads. Now toss it 100 times, and check again. Now try and do it 1000 times. Difficult right? Imagine tossing it a million times, how many times do you think it will land on heads. Let us find out using the concepts of probability of an event.

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Probability of an Event

Very simply put, a probability is the chance of something happening. Probability is a measure of the likelihood of a given event’s occurrence. There are events which we cannot predict with certainty, so we find out the probability of their occurrence. For example, when we say there is a 50% chance of rain today, we are calculating the probability of this event’s (rain) occurrence.

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 Probability of an Event

Formula of Probability

The most basic and general formula to calculate probability is

Probability = \( \frac{Number of favorable events}{(Total number of events)} \)

Terms related to Probability

  • Random Experiment: A random experiment is the one in which all the possible results are known in advance but none of them can be predicted with certainty.
  • Outcome: The result of a random experiment is called an outcome.
  • Sample Space: The set of all the possible outcomes of a random experiment is called a Sample Space, and it is denoted by  ‘S’.
  • Event: A subset of the sample space is called an Event.

Range of Probability

Range of Probability

If an event is impossible its probability is zero. Similarly, if an event is certain to occur, its probability is one. The probability of any event lies in between these values. It is called the range of probability and is denoted as 0 ≤ P (E) ≤ 1.

Finding the Probability of an Event

So let us explore this via an example. Say two coins are tossed, you must find the probability of the following events.

  1. At least one tails turns up
  2. No heads turn up
  3. At the most one tails turns up

So to find the probability of an event A in a finite space S.

P (A) = \( \frac{No. of sample points in A}{No. of sample points in S} \) = \( \frac{n(A)}{n(S)} \)

Here, S = { HH, HT. TH. TT }

  1. Here let A be event when at least one head turns up, So A = {HT, TH, TT} ; n(A) = 3 ; P (A) = \( \frac{n(A)}{n(S)} \) = 3/4
  2. Here A is the event where no heads turn up, so A= {TT} ; n(A) = 1 ; P (A) = \( \frac{n(A)}{n(S)} \) = 1/4
  3. Here A is the event when at the most one tails turns up, so A = {HH, TH, HT} ; n(A) = 3 ; P (A) = \( \frac{n(A)}{n(S)} \) = 3/4

Solved Examples for You

Question 1: Find the probability for a randomly chosen month to have its 10th day on a Sunday,

Answer : So lets solve this step by step

The probability of choosing any month from the given 12 months is 1/12

There are 7 possible days the 10th of a month can fall on. So the probability it falls on a Sunday is 1/7.

Thus the probability of a randomly chosen month to have its 10th day on a Sunday is

1/12 × 1/7 = 1/84

Question 2: One integer is chosen from 1,2,3…100. What is the probability that it is neither divisible by 4 nor divisible by 6?

Answer : From numbers 1-100

Numbers divisible by 4 = 25

Numbers divisible by 6 = 16

Numbers divisible by 12 (LCM of 4 and 6) = 8

Numbers divisible by 4 or 6 = 25 + 16 – 8 = 33

Numbers which are not divisible by 4 or 6 = 100-33 = 67

Required Probability = 67/100 = 0.67

Question 3: How to calculate the probability of an event?

Answer: For calculating the probability of an event, firstly divide the number of events by possible outcomes. By this, you will get the probability of a single event occurring. For example, for rolling dice for 5, the number of events is 1 (as there is only a single 5 on a dice) and the total number of outcomes is 6.

Question 4: What is an event in probability?

Answer: The set of outcomes from an experiment in probability is known as an event. For example, on tossing a coin there are two outcomes that are coin landing on ‘heads’ or ‘tails’. And these outcomes can be said to be the events connected with the experiment.

Question 5: How to find the probability of multiple events?

Answer: For this, multiply the probability of the first event by the probability of the second event. E.g., if the probability of happening of an event A is 3/9 and the probability of another event B is 2/9 then the probability of happening of both events at the same time is 3/9 × 2/9 = 6/81 = 2/27.

Question 6: What is a simple event probability?

Answer: It refers to the probability where a single event happens at a time and it will be having only a single outcome. Also, its probability lies between 0 and 1.

 

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