Using the Binomial distribution

Question

Using the Binomial distribution,

If n = 7 and p = 0.3, then calculate the following

P(x less than 3) =
[three decimal accuracy]
P(x at most 3) =
[three decimal accuracy]
P(x = 3) =
[three decimal accuracy]
P(x at least 3) =
[three decimal accuracy]
P(x greater than 3) =
[three decimal accuracy]

Answer 1

The probability mass function for the binomial distribution is:

P(X = k) = _nC_k p^k(1-p)^n-k,

where k represents the number of 'successes', n is the total number of trials, and p is the probability of success. In your case, p = 0.3, n = 7, and k will vary based on the problem.

_nC_k is a combination function that is often found on a scientific calculator, but can also be solved using:

n!/ (n-k)!k! . Rememer that factorials (!) just mean to do something like: 5! = 5*4*3*2*1 .

Given all of that, to set up your problems you have to make some decisions about how to do the least math. For example, for P(x < 3) we could choose to approach it directly and solve for P(x = 0) + P(x = 1) + P(x = 2) OR we could approach it indirectly and set it up as P(x < 3) = 1- P(x >= 3) and solve for

1 - [P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6) + P(x = 7)]. Clearly for this one, the direct approach is best because we only have to use the formula three times instead of 5.

So next, it's just plugging in the values we have into the formula:

P(x = 0) = ₇C₀(0.3)⁰(1 - 0.3)^7-0 = 0.082

P(x = 1) = ₇C₁(0.3)¹(1 - 0.3)^7-1 = 0.247

P(x = 2) = ₇C₂(0.3)²(1 - 0.3)^7-2 = 0.318

Then

P(x < 3) = P(x = 0) + P(x = 1) + P(x = 2)

so

P(x <3) = 0.082 + 0.247 + 0.318 = 0.647

The others can be set up accordingly.