Can Calculating Cumulative Binomial Probabilities Be Simplified?

[Mdhiggenz]

Homework Statement

Of all the weld failures in a certain assembly, 85%
of them occur in the weld metal itself, and the remaining
15% occur in the base metal. A sample of
20 weld failures is examined.

a. What is the probability that fewer than four of
them are base metal failures?

Is there a faster way to solve rather than doing p(x=0)+p(x=1)+p(x=2)+p(x=3)?

Thanks

Brandon

Homework Equations

The Attempt at a Solution

 

[Ray Vickson]

Homework Statement

Of all the weld failures in a certain assembly, 85%
of them occur in the weld metal itself, and the remaining
15% occur in the base metal. A sample of
20 weld failures is examined.

a. What is the probability that fewer than four of
them are base metal failures?

Is there a faster way to solve rather than doing p(x=0)+p(x=1)+p(x=2)+p(x=3)?

Thanks

Brandon

Homework Equations

The Attempt at a Solution

No, the way you said is about as short as possible. You can sometines speed things up a bit by doing it recursively: if
$$P(k) = {n \choose k} p^k (1-p)^{n-k}$$
we have
$$\frac{P(k+1)}{P(k)} = r(k) \equiv \frac{n-k}{k+1} \frac{p}{1-p},$$
so if we start from $P(0) = (1-p)^n$, we can get $P(1) = r(0) P(0),$ $P(2) = r(1) P(1),$ etc.