Visualizing the Binomial Distribution (6.6)

in this video we'll be learning about the binomial distribution before we talk about the binomial distribution let's review the binomial formula from the previous video we know that the binomial formula looks like this where k is the number of successes n is the number of trials and little p is the probability of success if we were flipping a coin two times and if a success is getting heads we can use this formula to calculate the probabilities for each number of successes this means that k will be equal to 0 1 or 2. in this case k cannot exceed 2 since we are only flipping the coin twice it would be impossible for us to get three successes in other words three heads by only flipping the coin twice n is equal to two because there are two trials and we know this because we are flipping the coin for a total of two times little p is equal to 0.5 because there is a 50 chance of getting heads when we flip the coin in other words there is a 50 chance of getting a success now that we have identified each variable we will use the formula to determine the probability of each value of k after plugging in the necessary values into the formula you should get 0.25 0.50 and 0.25 again as your final answers these are the probabilities for each respective value of k to visualize the binomial distribution we will create a bar chart with this data the number of successes is listed on the x-axis which is why we have 0 1 and 2. these are the values of k and the probability of success for each value of k is listed on the y axis so from this bar chart we can clearly see that the probability of getting 0 successes is 0.25 the probability of getting one success is 0.5 and the probability of getting two successes is 0.25 these are the probabilities that we just calculated using the binomial formula now let's see what happens if we increase the number of trials n if n is equal to 10 which means we are flipping the coin ten times instead of two times we can see that the shape of the binomial distribution begins to look like a normal distribution if you watched my previous videos you would have learned that the mean of the normal distribution is always located at the center of the distribution in this case we see that the mean of the binomial distribution is centered around 5. we can actually calculate the parameters of a binomial distribution if a variable x follows a binomial distribution the mean mu is equal to n times p the variance is equal to np times 1 minus p and the standard deviation sigma is just equal to the square root of the variance which is equal to the square root of np times 1 minus p we saw that as n increases the binomial distribution starts to look like the normal distribution now let's see what happens when we change the value of p here i have a bar graph with a little p of 0.5 and an n of 10. by looking at this distribution we can say that it is roughly normally distributed if we increase the value of p or decrease the value of p while keeping the value of n the same we see that the shape of the distribution changes as a result we can say that the shape of the binomial distribution depends on the value of p any deviations from 0.5 causes the binomial distribution to be skewed and this makes sense because if we look at the distribution to the left when p is equal to 0.1 this means that we only have a 10 chance of success this means that for the 10 trials or 10 attempts we should expect to get very little success which is why the probability is higher towards zero and less towards 10. with a probability of success of only 10 percent it is very unlikely for us to get a success 10 times in a row for each of the 10 trials which is why the probability of getting 10 successes is so small that we can't even see it on the bar graph in contrast when we increase the value of p as seen in the distribution to the right where the value of p is 0.8 we should expect to get a lot more successes since we have a higher success rate since the probability of success is 80 percent it makes sense that most of the data clumps towards the right side as shown in the bar graph we can also say that the data just generally clumps around the mean of the binomial distribution remember that the mean mu of a binomial distribution is equal to n times p if we make this calculation for each of these bar graphs we can clearly see that the data always clusters around the mean mu looking at the overall picture when p is equal to 0.5 it allows the distribution to be symmetrical a value of p less than 0.5 generally causes the distribution to be skewed to the right and a value of p greater than 0.5 generally causes the distribution to be skewed to the left with these fixed values of p the only way to cause the binomial distribution to not be skewed is by increasing the value of n as we mentioned before as the value of n increases the binomial distribution approaches the normal distribution when working with skewed distributions we have to significantly increase the value of n so that it becomes normally distributed however when we are working with symmetrical distributions we don't have to increase the value of n that much to approach normality since the distribution was already symmetrical to begin with because of this we can make a rough guideline we can assume a normal approximation of the binomial distribution if it follows two conditions the first condition is that n times p must be greater than or equal to 10. and the second condition is that n times 1 minus p is also greater than or equal to 10. both of these conditions must be satisfied in order for us to assume a normal approximation of the binomial distribution now i say this is a rough guideline because some people use 5 instead of 10 so you should check with your professor or textbook to see what guideline they are using so let's quickly recap what we've learned the value of p controls the shape of the distribution when the probability of success is equal to 0.5 it gives us a symmetrical distribution but when the probability of success deviates from 0.5 it gives us a skewed distribution a value of p greater than 0.5 tends to be skewed to the left and a value of p less than 0.5 tends to be skewed to the right as the value of n increases the binomial distribution approaches the normal distribution and finally if a variable x follows a binomial distribution then the mean mu is equal to np the variance is equal to np times 1 minus p and the standard deviation sigma is equal to the square root of np times 1 minus p if you found this video helpful consider supporting us on patreon to help us make more videos you can also visit our website at simple learning pro.com to get access to many study guides and practice questions thanks for watching

Channel: Simple Learning Pro

Views: 11,465

Keywords: Statistics (Field Of Study), Statistics, stats, ap stats, ap statistics, Lesson, Tutorial, explained, math, mathematics, Simple Learning Pro, Practice Questions, Binomial Distribution, Binomial Formula, visualizing the binomial distribution, visualizing

Id: Y2-vSWFmgyI

Length: 7min 21sec (441 seconds)

Published: Thu Jan 07 2021