# Probability question regarding bits used to test a micro-circuit

1. Nov 4, 2012

### wolfmanzak

1. The problem statement, all variables and given/known data

In a random pattern of eight bits used to test a micro-circuit, each bit is equally likely to be 0 or 1. Assume the values of the bits are independent.

a. What is the probability that all eight bits are 1?
b. What is the probability that exactly three of the bits are 1?
c. What is the probability that at least six of the bits are 1?
d. What is the probability that least two of the bits are 1?

2. Relevant equations

I'm not really sure, I'm trying to figure this part out. I've tried several different combinations of "formulas" for parts b-d. For b, I tried finding the probability of 3 bits being 1 and 5 being 0 but my calculations failed to yield a correct result. Any help in solving parts b-d is greatly appreciated. Thanks in advance.

3. The attempt at a solution
I've figured out part a. I used $$P(Bit)=0.5$$ because it is equally likely to be 0 or 1. And I used $$P(all = 1)=P(bit)^8=(0.5)^8=.003906$$.

2. Nov 4, 2012

### LCKurtz

Do you know the Binomial distribution? If the probability of success is p, then in n independent trials, if X is the number of successes then$$P(X = k) = \binom n k p^k(1-p)^{n-k}$$You can use that to answer your questions. For example, to verify your answer for all 8 bits = 1, you would calculate$$P(X=8) = \binom 8 8 \left(\frac 1 2\right)^8\left(1-\frac 1 2\right)^{8-8}$$which agrees with your result.