Calculating Probabilities

The Problem

Let X be a random variable following a Binomial distribution. All Binomial distributions have two parameters: number of trials and success probability. For X, n = 12 and p = 0.77. An example of where such a distribution may arise is the following:

You have a bag of candy made by Statistics, Inc. The company makes only two flavors: sour apple and anchovie. In its processing plant, 77% of the candy made is sour apple. Your bag holds 12 pieces. Define the random variable X as the number of sour apple pieces of candy in your bag.

For those who like pictures, here is a graphic of the probability mass function. Note that the distribution is skewed left:

Continuing the candy example, let us determine the probability that there are exactly 6 pieces of sour apple candy in the bag. In symbols, calculate P[X = 6].

Your Answer

In the box below, please enter the value of P[X = 6], where X ~ Binom(n=12, p=0.77), then click on the “Check your answer!” button. Please round your answer to the ten-thousandths place.

Your guess:

Another Binomial Distribution

Would you like to continue working on this topic? If so, click here for another Binomial distribution.

Assistance

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P[X = x]	the probability that the random variable takes on the value x

nCx	the number of combinations of n objects taken x at a time nCx = n!/(x!(n-x)!)

n	the number of trials
x	the stated number of successes
p	the success probability for each trial

P[X = x]	=	nCx × p^x × (1 − p)^{n − x}

	=	12C6 × 0.77⁶ × (1 − 0.77)^{12 − 6}

	=	924 × 0.208422380089 × 0.000148035889

	=	0.028509088907349

© Ole J. Forsberg, Ph.D. 2024. All rights reserved.		.