What is the probability of at least three successes in six trials of a binomial experiment in which the probability of success is 50%?

Answer 1

Hi Emily,

The formula for counting x successes in n trials, where p is the chance of success (.50) and q = 1-p = .50 is the chance of failure, is:

(nCx) * p^x * q^n-x

nCx can usually be done in a calculator, please message me if you need more help on that.

Since we’re interested in “at least three” successes then we’ll need to find several probabilities and add them up:

Chance of exactly 3 successes: 6C3 * (.50)³ * (.50)³ = .3125

4 successes: 6C4 * (.50)⁴ * (.50)² = .2344

5 successes: 6C5 * (.50)⁵ * (.50)¹ = .0938

6 successes: 6C6 * (.50)⁶ * (.50)⁰ = .0156

Total probability is .6563 or 65.63%.

Please let me know if I can be more helpful, thanks!

Joe

Answer 2

the binomial formula is _nC_r(p)^r(q)^n-r where n = number or trials, r = number of successes

with p = probability of success and q = probability of failure (C denotes combination)

Therefore, at least 3 success means either 3 or 4 or 5 or 6 successes.with addition of the combinations of each.

₆C₃(.5)³(.5)³ + ₆C₄(.5)⁴(.5)² + ₆C₅(.5)⁵(.5)¹ + ₆C₆(.5)⁶(.5)⁰ = .3125 + .2344 + .0938 + .0156 = .6563