n=10, p=0.15, x < or = 4. prob. of x < or = 4 successes is ___ Round to 4 dec. pts

Question

A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment, n=10, p=0.15, x < or = 4. The probability of x < or = 4 successes is ___. Round to four decimal places if needed

Answer 1

If p=0.15, then q=1-p=0.85, the probability of failure, then we have the binomial distribution:

(p+q)ⁿ=pⁿ+np^n-1q+n(n-1)p^n-2q²/2!+... where n=the number of independent trials=10. The sum of these terms =1 because (p+1-p)ⁿ=1 for all n.

Each term in this series represents an individual exact probability:

(10 successes)+(9 successes)+...+(4 successes)+(3 successes)+(2 successes)+(1 success)+(no successes). Alternatively, this is (10 failures)+(9 failures)+...+(1 failure)+(no failures).

x≤4 successes means at least 6 failures=P(q=10)+P(q=9)+P(q=8)+P(q=7)+P(q=6).

(q+p)ⁿ=qⁿ+nq^n-1p+n(n-1)q^n-2p²/2!+...=

0.85¹⁰+10(0.85⁹)(0.15)+45(0.85⁸)(0.15²)+120(0.85⁷)(0.15³)+210(0.85⁶)(0.15⁴)=0.9901.