Binomial Distribution

Recall that for a binomial distribution, we must have: Two possible outcomes, called success and failure Constant probability Independent trials A fixed number of trials

Ex 1: Consider the random event with two possible outcomes, such as heads and tails, from tossing a coin. Each toss is independent of every other toss and probability is constant. Suppose we define X to be the number of heads in a series of 4 tosses. What possible values can X take on? X can be {0,1,2,3,4}.

X P(X) We can find the probabilities by listing the possible outcomes and counting: HHHHHHHTHHTH HHTT HTHHHTHTHTTH HTTT THHHTHHTTHTH THTT TTHHTTHTTTTH TTTT The binomial formula is: We can confirm the probabilities found above /164/166/164/161/16

We can find the probabilities by listing the possible outcomes and counting: The binomial formula is: We can confirm the probabilities found above. HHHHHHHTHHTHHHTT HTHHHTHTHTTHHTTT THHHTHHTTHTHTHTT TTHHTTHTTTTHTTTT X01234 P(X)1/164/166/164/161/16

One thing that makes the binomial formula so nice to use is that the graphing calculator has this formula programmed already! Let’s try this on the calculator. Enter {0,1,2,3,4} into L 1. You can do this from the homescreen or from within lists. Enter the probabilities in L 2 using the command binompdf(4,0.5, L 1 ).

The values are the same as we found from listing the outcomes or using the formula by hand. We can use this same process to find a single value. To find the binomial probability for n = 6, p = 0.2, X = 3, use the command binompdf(6,.2,3). The probability is

You should know how to use the formula, but as a practical matter, the calculator is fine. See that we can make a single calculation readily using binompdf. Pdf stands for probability density function. When dealing with discrete distributions like the binomial, a calculation of a pdf will give us a probability.

When working with a continuous distribution, such as the normal distribution, we used a cdf or continuous density function, to find an area under the curve. We can also use a cdf with the binomial distribution, but the calculator defines it a little differently. It sums all the probabilities for X = 0 up to X = the given value of X. If you want to find the probability for having up to 4 girls in a family of 5 children, we have n = 5, p =.5, X ≤ 4. Try the command binomcdf(5,.5,4). P(X ≤ 4) =.9687.

Going back to the earlier example using the coin toss, check your calculator. Have {0,1,2,3,4} in L 1 and the binompdf values in L 2. Now enter binomcdf(4,.5,L 1 ) into L 3. Now let’s look at the lists. The values of X are in L 1. The probability of each vales is in L 2. Now the sum of each probability (a cumulative sum) is in L 3. In fact, the cumSum(L 2 ) command will give the same values as the binomcdf.

When working a binomial problem I usually suggest that students make a simple number line to aid in finding which values to include. It sounds unnecessary, but really it is often needed to avoid mistakes. Ex 2: Find the probability of getting at most 4 out of 10 true and false problems correct by guessing (assume T and F are equally likely). Answer: The possible values of X are 0,1,2,3,…, “At most 4” tells us that we could have {0,1,2,3,4}. Use the command binomcdf(10,.5,4)=.3769.

Ex 3: Find the probability of making 5 or more baskets out of 9 shots, for an 80% free throw shooter in basketball. Answer: The possible values of X are {0,1,2,3,…,9} “5 or more” tells us that we could have {5,6,7,8,9}. To find this quantity we use the binomial command on the calculator. 1- binomcdf(9,.8,4)=.9804.

Nex 4: Find the probability of missing 5 or more baskets out of 9 shots, if you are an 80% free throw shooter in basketball. Answer: The possible values of X are {0,1,2,3,…,9} Missing 5 or more means hitting only 0,1,2,3,or 4. To find this quantity we use binomcdf(9,.8,4)=

You are now ready to work problems on your own. Make sure to draw numberlines whenever there are inequalities. You’d be surprised at how easy it is to make a mistake without the numberline.