I am having some trouble with statistics and there isn't anything in my book to help me.

A Binary String is a sequence of digits either 0 or 1.

a.) How many binary strings of length 9 are there? (The first digit may be zero).

b.) How many binary strings of length 9 contain the digit 1 exactly 3 times?

c.) How many binary strings of length 9 alternate 0’s and 1’s?

d.) How many binary strings of length 9 have an even number of 1’s?

a) Imagine you are told to draw a probability tree for that. I hope you are quite familiar with probability trees.

On the first digit, you have either 0 or 1. (2 possibilities or 2^1)
Upon the second digit, you have the sequence becoming 00, 01, 10 or 11. (4 possibilities or 2^2)
Upon the third digit, you have the sequence becoming 000, 001, 010, 011, 100, 101, 110 or 111. (8 possibilities or 2^3)
Do you find a sequence here? Each time, the number of possibilities is doubled. Hence, you can say that for a sequence of length 10 has 2^5 = 32 possibilities.

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b) Here, you start with the Binomial Distribution.

Let X be the probability that a one is in the sequence.
X ~ B(0.5, 9)

[0.5 is the probability of having a 1 when the length of the sequence is 1 and 9 is the length of the sequence]

So, P(X = 3) = 9C3(1-0.5)^{9-3}(0.5)^3

You can do the math.

c) I'm not sure of this one, but alternate 0's and 1's means either 010101010 or 101010101. So I see only 2. The probability of having that is 2/[Total number of binary strings]

d) This one, you can find it using:

P(X = 2) + P(X = 4) + P(X = 6) + P(X = 8)

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