In a test of a printed circuit board using a random test

Answer

Step 1 of 3

(a)

In a test of a printed circuit board using a random test pattern, an array of  bits is equally likely to be  or

Assume the bits are independent.

we are asked to find the probability that all bits are

Let’s create the sample space of bits.

We can see that there are two outcomes in sample space, whether it can be  bit or  bit.

Hence the probability of occurring 1 bit is

Similarly, the probability of occurring 0 bit is

We have given in the question that our bits are independent.

Hence we can apply the independence rule of probability which is,

If two events  and  are independent, then we can write,

Similarly, for multiple events  are independent if and only if for any subset of these events,

……(1)

Let  denote the event that the  bit is a one.

Then by independence rule, we can write the probability that all bits are  using equation (1),

………(2)

We know the probability of occurring 1 bit is

Now we can write the equation (2),

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Hence the probability that all bits are  is