Question

Your DNA code is composed of a series of four nucleotides: adenine, guanine, thymidine and cytosine (A, G, T and C, respectively).

1) What is the probability an individual has the following nucleotide sequence: “TATATA” at any particular position? (assuming independence).

2) What is the probability that an individual has k T’s in their DNA code at any particular position? (k can be any integer and you may assume independence). Here we're looking for the probability of k consecutive T's.

Answer 1

Answer on Question #43187-Math-Statistics and Probability

Your DNA code is composed of a series of four nucleotides: adenine, guanine, thymidine and cytosine (A, G, T and C, respectively).

1) What is the probability an individual has the following nucleotide sequence: "TATATA" at any particular position? (assuming independence).

2) What is the probability that an individual has k T's in their DNA code at any particular position? (k can be any integer and you may assume independence). Here we're looking for the probability of k consecutive T's.

Solution

1) The probability an individual has the following nucleotide sequence: "TATATA" at any particular position is

$P(T \text{ is first} \& A \text{ is second} \& T \text{ is third} \& A \text{ is fourth} \& T \text{ is fifth} \& A \text{ is sixth}).$

And since "&" tells us to multiply probabilities:

$P(T \text{ is first}) \cdot P(A \text{ is second}) \cdot P(T \text{ is third}) \cdot P(A \text{ is fourth}) \cdot P(T \text{ is fifth}) \cdot P(A \text{ is sixth}).$

The probability of A or T is

$$P(A) = P(T) = \frac{1}{4}.$$

So, the probability an individual has the following nucleotide sequence: "TATATA" at any particular position is

$$P(\text{TATATA}) = \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} = \frac{1}{4^6} = \frac{1}{4096}.$$

2) Therefore the probability of k consecutive T's

$$P\left(\frac{k}{\text{TTT} \dots \text{T}}\right) = \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot \dots \cdot \frac{1}{4} = \frac{1}{4^k}.$$

Answer: 1) $\frac{1}{4096}$; 2) $\frac{1}{4^k}$.

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