this lesson will show how to use desmos comm to graph a binomial distribution and determine probability let's begin by discussing the characteristics of a binomial experiment number one there are a fixed number of trials the trials of the repeated experiment the variable n denotes a number of trials number two there are only two possible outcomes for each trial success or failure the variable P denotes the probability of a success on one trial and Q often denotes a probability of a failure on one trial and therefore P plus Q must equal 1 or 100% number 3 the n trials are independent and are repeated using identical conditions because the end trials are independent the outcome of one trial does not affect the outcome of another trial and then finally number 4 the random variable x equals the number of successes in n independent trials let's look at an example a fair coin is tossed 50 times because the coin is fair there's a 50% chance or 1/2 chance of getting a head and a 50% chance or 1/2 chance of getting a tail we're asked to determine each probability and give the answer as a decimal three decimal places and a percent to one decimal place let's begin by graphing the binomial distribution using decimals to begin we go to desmos comm and click graphing calculator in cell 1 we can either type in binomial district functions click distribution and click binomial distances prompting us to now enter the number of trials followed by the success probability so the coin is tossed 50 times and therefore there are 50 trials , the coin is fair and therefore there's a 50% chance of getting a head and therefore the success probability is 50% 0.5 or 1/2 I will enter 0.5 let's close the desmos keypad right now we do not have a good window to view movie graph to quickly adjust the window we click the zoom fit button below the red circle which is the magnifying glass with a plus and now we have a nice graph of the binomial distribution we want to we can change the color by clicking and holding on the red circle and selecting a different color I'm going to go ahead and select blue to close this menu click outside the menu now from here if we want to we can click on any of the points on the graph it will give us the corresponding probability for getting that many heads out of the 50 trials so for example here this ordered pair tells us the probability of getting 21 heads is approximately 0.060 or 6% and we can click on any of the points that we want let's go ahead and clear these but our first probability is the probability of getting 30 or more heads out of the 50 trials so we're gonna go ahead and click on this box here that says find cumulative probability and because we want the probability of 30 or more heads this would be the same as getting from 3 to 50 heads so we'll enter 30 for the min tab 50 for the max and the probability appears below the probability of getting 30 or more heads is approximately zero point one zero one or 10.1% so look at the graph for a moment notice how the closed or solid points represent all the successful outcomes of getting 30 31 32 33 and so on all the way out to 50 heads from the 50 trials let's go and record this and find our second probability so as a decimal we had approximately zero point 1 0 1 which is equal to 10.1% next we want the probability of 28 or less heads 28 or less would be from 0 to 28 so we change the min to 0 tab max to 28 we want 28 not 18 and the probability appears below three decimal places we have approximately zero point eight three nine or 83.9% again looking at the graph notice how the closed points and the vertical line segments represent the successes of getting 28 27 26 25 and so on all the way down to 0 heads so if we were to find all these individual probabilities and then find the sum of the probabilities we know it would be this value here of approximately zero point eight three nine again let's go ahead and record this probability approximately zero point eight three nine which is equal to 80 three point nine percent next we want the probability of getting from 25 to 35 heads including 25 and including 35 so now the men is 25 the max is 35 and the probability of getting from 25 to 35 heads out of 50 trials is approximately zero point five five five or fifty five point five percent and for number four we want the probability of getting exactly 30 heads to find this probability we can set both the min and Max to 30 which gives us approximately zero point zero four two or four point two percent also notice how if we click on the close point on the graph we do get the same result and zero point zero four two is equal to four point two percent I hope you found this helpful