Calculating Cumulative Probabilities

In the box below, please enter the value of P[X ≤ 1], where X ~ Binom(n=11, p=0.45), then click on the “Check your answer!” button. Please round your answer to the ten-thousandths place.

Your guess:

You got the correct answer of P[X ≤ 1] = 0.0139. Congratulations!

Unfortunately, your answer was not correct. Either try again or click on “Show Solution” below to see how to obtain the correct answer.

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For the Binomial distribution, the formula to calculate probabilities of single events is

For the Binomial distribution, there is no short formula for calculating cumulative probabilities. Thus, to calculate P[X ≤ x], one would need to calculate

Where the summation (Σ) happens over all values in the sample space of X that satisfy the requirement X ≤ x. In this formula, there are a few symbols to know:

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Thus, the probability that there is no more than one piece of sour apple candy in your bag of 11 pieces of candy is approximately 1.3931%.

Show the R Code

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Copy and paste the following code into your R script window, then run it from there.

In the R output, the number given after running the final line above is the probability requested, P[X ≤ 1]. There are three slots in the function. The first is x; the second, n; the third, p.

Show the Excel Formula

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Copy and paste the following code into an Excel cell, then hit enter.

The first slot is the value of x. The second slot is the value of n, the number of trials. The third slot is the value of p, the success probability. The last slot is FALSE for calculating P[X = x] and TRUE for calculating P[X ≤ x].