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Normal Approximation To The Binomial Distribution - Approximating A Binomial Probability
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In this video we discuss how and when to use a normal approximation to a binomial distribution. We go through the procedures as well as using a correction for continuity in the process.
Transcript/notes (partial)
The binomial distribution, which is a discrete probability distribution, is used to find probabilities in which there are only 2 outcomes, or can be reduced to 2 outcomes. For instance flipping a coin has only 2 outcomes, heads or tails, and on a test, a multiple choice question can be reduced to correct or incorrect.
And real quick there are 4 requirements for a binomial experiment. (Fixed number of trials, only 2 outcomes, independent, same probability for each trial).
Binomial probability is the probability of x successes in n trails.
The binomial formula, or the formula for getting exactly x successes in n trials is probability of x equals, n factorial divided by n minus x factorial times x factorial times p to the x times q to the n minus x. In the formula, probability of x, is the probability of x successes, n equals the number of trails, p equals the probability of success in an individual trial, and, q equals the probability of a failure in a single trial, which is 1 minus p. And the mean is equal to n times p and the standard deviation is equal to the square root of n times p times q.
This issue with using this formula is when n, the number of trails becomes large, the calculations can become very difficult, for instance, probability of 59 successes in 200 trials.
This is where, under certain conditions we can use a normal distribution as an approximation to calculate probabilities.
These conditions are. Number 1, the problem must meet the 4 requirements of a binomial distribution, number 2 is that n times p must be greater than or equal to 5, and n times q must also be greater than or equal to 5. So, for instance if n, the number of trials is 12 and p, the probability of success is 0.3, which equals 3.6, a normal distribution should not be used as an approximation.
And number 3 is that if conditions 1 and 2 are met, a correction for continuity must be applied. And this means that for any value for x, we must use boundaries. For instance, probability of 59 successes, which is probability of x = 59. In this instance, we would use probability of 58.5 less than x less than 59.5. So, 58.5 and 59.5 would be our boundaries. Below is a table listing the different probability scenarios of when to add or subtract 0.5, and we are going to go through several of these using examples.
Let’s say that a certain golfer lands his ball in the fairway 72% of the time using his driver. 3 part question. Number 1 is what is the probability that he hits 43 fairway drives in 54 total driver shots? Number 2, what is the probability he hits at most 39 fairway drives in 54 total driver shots? And number three, what is the probability that he hits more than 41 fairway drives in 54 total driver shots?
In this situation, n, the number of trials = 54, p, the probability of success is 72% or 0.72, and q = 0.28, which is 1 minus p. And the mean is n times p, 54 times 0.72, which equals 38.88, and the standard deviation is equal to the square root of n times p times q, square root of 54 times 0.72 times 0.28, which equals 3.299.
Before we do any calculating we first need to make sure the question meets the conditions to use a normal distribution as an approximation.
It does meet the 4 requirements for a binomial experiment. Since conditions 1 and 2 are met, we need to apply the correction for continuity for each of the 3 different questions.
So, for question 1, we are looking for the probability that x = 43, and in the table, when x is equal to a value, we subtract 0.5 and add 0.5 to the value to get the boundary. So, on a normal distribution we are looking for the probability of x less than 39.5.
For question 2, we are looking for the probability of at most 39, so, 39 is included, so we want the probability that x is less than or equal to 39, and in the table, when x is less than or equal to a value, we add 0.5 to the value to get the boundary. So, on a normal distribution we are looking for the probability of x less than 39.5.
And for question 3, we are looking for the probability of more than 41, so, 41 is not included, so we want the probability that x is greater than 41, and in the table, when x is greater than a value, we add 0.5 to the value to get the boundary. So, on a normal distribution we are looking for the probability of x greater than 41.5.
Timestamps
0:00 What Is The Binomial Distribution?
0:18 4 Requirements For A Binomial Distribution
0:50 Formula For Binomial Distribution
1:52 Conditions For Using A Normal Approximation To A Binomial Distribution
2:24 Correction For Continuity
2:58 Example Problem For Normal Approximation To A Binomial Distribution
5:24 Calculating Probabilities For Normal Approximation To A Binomial Distribution
Transcript/notes (partial)
The binomial distribution, which is a discrete probability distribution, is used to find probabilities in which there are only 2 outcomes, or can be reduced to 2 outcomes. For instance flipping a coin has only 2 outcomes, heads or tails, and on a test, a multiple choice question can be reduced to correct or incorrect.
And real quick there are 4 requirements for a binomial experiment. (Fixed number of trials, only 2 outcomes, independent, same probability for each trial).
Binomial probability is the probability of x successes in n trails.
The binomial formula, or the formula for getting exactly x successes in n trials is probability of x equals, n factorial divided by n minus x factorial times x factorial times p to the x times q to the n minus x. In the formula, probability of x, is the probability of x successes, n equals the number of trails, p equals the probability of success in an individual trial, and, q equals the probability of a failure in a single trial, which is 1 minus p. And the mean is equal to n times p and the standard deviation is equal to the square root of n times p times q.
This issue with using this formula is when n, the number of trails becomes large, the calculations can become very difficult, for instance, probability of 59 successes in 200 trials.
This is where, under certain conditions we can use a normal distribution as an approximation to calculate probabilities.
These conditions are. Number 1, the problem must meet the 4 requirements of a binomial distribution, number 2 is that n times p must be greater than or equal to 5, and n times q must also be greater than or equal to 5. So, for instance if n, the number of trials is 12 and p, the probability of success is 0.3, which equals 3.6, a normal distribution should not be used as an approximation.
And number 3 is that if conditions 1 and 2 are met, a correction for continuity must be applied. And this means that for any value for x, we must use boundaries. For instance, probability of 59 successes, which is probability of x = 59. In this instance, we would use probability of 58.5 less than x less than 59.5. So, 58.5 and 59.5 would be our boundaries. Below is a table listing the different probability scenarios of when to add or subtract 0.5, and we are going to go through several of these using examples.
Let’s say that a certain golfer lands his ball in the fairway 72% of the time using his driver. 3 part question. Number 1 is what is the probability that he hits 43 fairway drives in 54 total driver shots? Number 2, what is the probability he hits at most 39 fairway drives in 54 total driver shots? And number three, what is the probability that he hits more than 41 fairway drives in 54 total driver shots?
In this situation, n, the number of trials = 54, p, the probability of success is 72% or 0.72, and q = 0.28, which is 1 minus p. And the mean is n times p, 54 times 0.72, which equals 38.88, and the standard deviation is equal to the square root of n times p times q, square root of 54 times 0.72 times 0.28, which equals 3.299.
Before we do any calculating we first need to make sure the question meets the conditions to use a normal distribution as an approximation.
It does meet the 4 requirements for a binomial experiment. Since conditions 1 and 2 are met, we need to apply the correction for continuity for each of the 3 different questions.
So, for question 1, we are looking for the probability that x = 43, and in the table, when x is equal to a value, we subtract 0.5 and add 0.5 to the value to get the boundary. So, on a normal distribution we are looking for the probability of x less than 39.5.
For question 2, we are looking for the probability of at most 39, so, 39 is included, so we want the probability that x is less than or equal to 39, and in the table, when x is less than or equal to a value, we add 0.5 to the value to get the boundary. So, on a normal distribution we are looking for the probability of x less than 39.5.
And for question 3, we are looking for the probability of more than 41, so, 41 is not included, so we want the probability that x is greater than 41, and in the table, when x is greater than a value, we add 0.5 to the value to get the boundary. So, on a normal distribution we are looking for the probability of x greater than 41.5.
Timestamps
0:00 What Is The Binomial Distribution?
0:18 4 Requirements For A Binomial Distribution
0:50 Formula For Binomial Distribution
1:52 Conditions For Using A Normal Approximation To A Binomial Distribution
2:24 Correction For Continuity
2:58 Example Problem For Normal Approximation To A Binomial Distribution
5:24 Calculating Probabilities For Normal Approximation To A Binomial Distribution
0:14:10
The Normal Approximation to the Binomial Distribution
0:08:01
Normal Approximation To The Binomial Distribution - Approximating A Binomial Probability
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