Cumulative Binomial Probabilities Assignment Help

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A classic illustration of cumulative binomial probability is tossing a coin n times. Let’s say, we toss a coin 10 times. We want to see how many “heads” occur.

Then, X = the random variable =Number of heads out of 10 flips done consecutively.

Here, the random variable X∼Binomial(n=10, p=1/2).

Now, if we have to find, how many heads can occur out of 10 random flips or what is the probability of getting 5 heads out of 10 flips?

We could answer this using the pmf:

Pr{X=5} = (10/5) (1/2)⁵(1 – ½)⁵=

252∗ (0.03125) ∗ (0.03125) = 0.2460938

So, the probability of getting exactly 5 times head out of 10 random flips is 0.2460924

Now, if we ask the question: what is the probability of getting 5 or fewer heads? We have to undertake the process of finding the cumulative binomial probability.

We have to use the following formula for the same:

P_r{X≤5} =∑k=15(10k) (1/2) k (1−1/2)10−k

Where, k = the probability of occurring head which will be 5,4,3,2,1,0.

= (0.5) (0.0009765625) +10∗ (0.5) (0.001953125) +45(0.25) (0.00390625) +120(0.125) (0.0078125) +210(0.0625) (0.015625) +252(0.03125) (0.03125)

=0.6230469

As, the probability of k changes but accumulates till it becomes zero, so, the process becomes cumulative type.

Cumulative Binomial Probabilities have wide many applications like the example given above. It provides information on existence or non-existence of an event within a certain sample.

Suppose in a population 30% people have no health insurance. Now, if a sample of 20 people (X= 20) is taken and we are asked to find what is the probability that more than 9 have no health insurance then we have to use the process of cumulative binomial probabilities.

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