Binomial Distribution in Probability
In our everyday life, frequently we are engaged in computing the probability of the desired outcome. Be it whether India will win a test series or the price of a stock will rise or fall in the near future, whether it will rain on a particular day or how much a chance does a student stand to crack a particular exam. We are surrounded by a world fuelled by the study of probabilities. The probability of an event happening helps one to be better prepared thereby aiding in improved decision making. The occurrence or failure of an event cannot be predetermined as it depends on varied parameters but knowing the probability of its happening or in fact not happening does simplify things as we are better prepared for the eventualities.
In this blog, we shall deal with one very important tool and aspect of probability that helps in solving a good many questions known as Binomial Distribution. Problems on this part are quite common in competitive exams and not knowing it will lead to losing some very crucial marks.
Suppose there are n independent trials of a random experiment; be it tossing an unbiased coin or picking a card from a well-shuffled card. Let’s say r be the number of times an event A occurs. Let’s say the probability of A occurring has the same probability P(A) = p and we refer to it as a success. The non-occurrence of A is called failure and its probability is given by q = 1-p.
Now, if the experiment is performed n times, then r is the number of times that A has occurred and (n-r) is the number of times it has failed to occur.
This is because A is happening r times and A not happening is n-r times. So, we need to pick r places out of n where A will occur and this can be done in ways and automatically A not happening will occur in rest of n-r places.
Let’s look at an example. We have an unbiased coin and it is tossed 10 times. Now the probability of head showing in any trial is ½ and the probability of tail showing is 1/2. If we are interested in knowing the probability that head will show exactly 6 times, then it also implies that tail has occurred in 4 of the trails. Thus, the probability of head occurring 6 times out of 10 times will be given by
Example 1: An unbiased coin is tossed 10 times. What is the probability of getting at least 7 tails?
Solution: Here n = 10, p = ½ and q = ½
Probability of getting at least 7 tails = Probability of getting exactly 7 tails + Probability of getting exactly 8 tails + Probability of getting exactly 9 tails + Probability of getting exactly 10 tails.
Example 2: From a set of two-digit numbers 10 to 99, four numbers are chosen one by one with replacement. An event A occurs if and only if the product of the two digits of a selected number is 18. What is the probability that A occurs at least 3 times?
Solution: Total number of two digits = 90
The numbers whose two digits’ product is 18 are 29, 36, 63 and 92.
Thus, the probability of A happening = 4/90 = 2/45 and this remains the same in all the four trials as the numbers are being replaced.
Now the probability that A happens at least 3 times = Probability that A happens exactly 3 times + Probability the A happens exactly 4 times.
Example 3: There are one hundred identical coins and each coin has a probability of p, of showing up heads. If the probability of heads showing on 50 coins is equal to that of the heads showing in 51 coins, then what is the value of p if 0 < p < 1 ?
Example 4: Six throws are made with a pair of dice. What is the probability of throwing doublets at least four times?
Solution: The possible cases of doublets are (1,1), (2,2), (3,3), (4,4), (5,5) and (6,6).
Total possible cases = 6*6 = 36
The probability of obtaining a doublet in a single throw = 6/36 = 1/6
The probability of throwing a doublet at least four times
I hope now you can solve the questions on Binomial Distribution quite easily. Practice some more questions to get a hang of this topic.