Let S be the set of all sequences of 6 zeros and 4 ones. Find the probability that a randomly selected element of S has no two consecutive ones. The sample space is 10!

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- Dec 9th 2011, 09:19 AMJskidprobability that binary number has no consecutive ones
Let S be the set of all sequences of 6 zeros and 4 ones. Find the probability that a randomly selected element of S has no two consecutive ones. The sample space is 10!

- Dec 9th 2011, 09:28 AMPlatoRe: probability that binary number has no consecutive ones
- Dec 9th 2011, 09:40 PMSorobanRe: probability that binary number has no consecutive ones
Hello, Jskid!

Quote:

Let be the set of all sequences of 6 zeros and 4 ones.

Find the probability that a randomly selected element of S has no two consecutive ones.

The sample space is 10! . No!

That would be true if we had 10*different*objects to arrange.

Since we have six identical 0's and four identical 1's,

. . the sample space is: .

Plato has the best solution.

I went about it differently ... and found that it required far more work.

Instinctively, I set about separating the 1's.

I arranged the four 1's in a row with a space before, after, and between them.

. .

Then I placed 0's in the three interior space:

. .

Now I must distribute the three remaining 0's.

They can all go into any of the 5 spaces: 5 ways.

Two can go into one space, one into another: . ways.

The three can go into separate spaces: . ways.

Hence, there are:. ways with no consecutive 1's.

I'm gratified that I got Plato's answer . . .

.