In simple words, the probability is the study of the possibility of occurrence of a particular event with respect to a number of other possible events, among which no two or more of them can occur simultaneously, that is at a given point in time only one of the possible events can occur. The simplest example of the application of probability is to determine the possibility of occurrence of a head that is ½, viz. favorable outcomes (occurrence of head) divided by possible outcomes (head and tail). Binomial distributionLet’s proceed with the same activity of tossing a coin, now suppose your friend suggests you throw a coin 3 times, and if a head appears at least once you have to throw him a treat. But, you know that you are left with very little money to spend. Here it becomes necessary for you to get an idea as to what would be the probability of you being forced to throw a treat. In such cases, where success and failure are involved in independent trials, the process used to calculate the probability is known as the binomial distribution method or the Bernoulli Distribution method. It has got such a name because it involves only two possible events that are success and failure that are distributed throughout the trials executed.
These points if fulfilled together, allow the usage of the Binomial Distribution method. Now getting back to the conditions set by your friend, we can see that all the required conditions to apply the binomial distribution method are fulfilled. This was a practical problem, now let’s work out and solve a theoretical problem on binomial distribution to understand it better. Let’s suppose there is an unbiased six-faced dice and to find the probability of rolling a number less than 3 at least 3 times within a total of 5 rolls of the dice. Basic Principle of arrangement For a better understanding of questions on the binomial distribution, it is important to understand the concept of arranging things in a row. There may be a number of ways to arrange identical or non-identical things, sometimes counting these arrangements becomes important. If there are (n + m) things out of which n are identical of one kind and m are identical of some other kind, then the total ways of arranging them in a row are given by (n + m)!/(m! × n!). Solution:
Sample Examples Question1: A coin is tossed 3 times, what is the probability that the tail will appear only once? Solution:
Question2: A person can hit a bullseye with a probability of 20%. What is the probability that he can hit it at most 3 times if 4 trials are given? Solution:
Question3: Two friends Rakesh and Simi were throwing unbiased dice one after the other. They have decided to stop only when one of them got an even number. Can the probability of Simi winning the game be obtained by the Binomial Distribution method? Why/Why not? Solution:
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A single six-sided die is rolled. Find the probability of rolling an even number or a number less than 5. What is the probability. |