## Binomial Probabilities

These kinds of questions can be answered using the binomial probability distribution:

1. If I flip a coin 20 times, what is the probability of getting 15 or more heads?

2. If I have a multiple choice test consisting of 12 questions, and each question has 4 answer choices, and I just guess at the answers, what is the probability of passing the test (i.e., getting at least 60% of them correct)?

Here is the formula for determining the probability of events with a binomial outcome (i.e., two possible outcomes --> heads/tails, or right/wrong, etc.):

p(r) = nCr * p^r * q^(n-r)

where:
r = the number of events of interest (i.e., 15 heads)
nCr = combinations of n, taken r at a time; or how many ways can we end up with r events in n trials
nCr = n! / r!(n-r)!
p = probability of success --> p(heads) = 0.5
q = probability of failure (1-p) --> 1-0.5 = 0.5

For question 1, we need to find the probability of 15 or more heads when we flip a coin 20 times, so we need to solve the binomial formula for r = 15,16,17,18,19, and 20 and add up the probabilities of each.

p(15) = 20C15 * 0.5^15 * 0.5^5 = 15504 * 0.0000305 * 0.03125 = 0.0148

Now, do the same thing for r=16 ,17,18,19,20 and add up all of them, and you have your answer.

Answer = 0.0207
So, in other words, if we flip a coin 20 times, the probability of getting 15 or more heads is just over 2%.

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Now, for problem 2, the idea is exactly the same, except that:

p = 0.25 (we have 4 answer choices, and if we just merely guess, the chances of getting it right are 1/4)

q = 1-p = 1-0.25 = 0.75

In order to pass the test, we need to get at least 60% right, which is 12 * 0.6 = 7.2, so we need to get at least 8 correct.

p(r) = nCr * p^r * q^(n-r)

r = 8,9,10,11,12
n = 12

p = 0.25
q = 0.75

p(8) = 12C8 * 0.25^8 * 0.75^4 = 495 * 0.0000153 * 0.3164 = .00239

Now, do the same thing for r=9,10,11,12 and add up all of the probabilities.

Answer = 0.0028
So, in other words, if we just take wild guesses, the probability of passing this particular test (12 questions, 4 choices per question) is around 0.28%, or 1 in 357.