# 10 different kinds of amino acids, which we will call A,B, C, D, E, F, G, H, I, and J. Charles creates a protein consisting of a sequence offour amino acids by first randomly selecting a type of amino acid for the first position,then randomly selecting a type of amino acid for the second position, then randomlyselecting a type of amino acid for the third position, and finally randomly selecting atype of amino acid for the fourth position. For each position in the sequence, each ofthe ten types of amino acids is equally likely to be selected. A type of amino acid canappear at more than one position in the sequence, so a sequence like (B, J, A, B) ispossible.Luis has four amino acids, and wants to make a copy of Charles’ sequence. Luiscan do this if any permutation of his amino acids matches Charles’ sequence.What is the probability that Luis can match Charles’ sequence, if Luis’ amino acidsare:(a) G, H, I, J(b) G, H, I, I(c) H, H, I, I(d) H, I, I, IExplain why you use each formula. Show your calculations. Express all answers to thenearest ten-thousandth.

Question

10 different kinds of amino acids, which we will call A,
B, C, D, E, F, G, H, I, and J. Charles creates a protein consisting of a sequence of
four amino acids by first randomly selecting a type of amino acid for the first position,
then randomly selecting a type of amino acid for the second position, then randomly
selecting a type of amino acid for the third position, and finally randomly selecting a
type of amino acid for the fourth position. For each position in the sequence, each of
the ten types of amino acids is equally likely to be selected. A type of amino acid can
appear at more than one position in the sequence, so a sequence like (B, J, A, B) is
possible.
Luis has four amino acids, and wants to make a copy of Charles’ sequence. Luis
can do this if any permutation of his amino acids matches Charles’ sequence.
What is the probability that Luis can match Charles’ sequence, if Luis’ amino acids
are:
(a) G, H, I, J
(b) G, H, I, I
(c) H, H, I, I
(d) H, I, I, I
Explain why you use each formula. Show your calculations. Express all answers to the
nearest ten-thousandth.

Step 1

Total number of different kinds of amino acids available for Charles= 10

As the creation of protein requires sequence of 4 amino acids and the repetition of amino acids are allowed, the total possible ways of creating a protein is,

Total possible sequences for Charles = 10 x 10 x 10 x 10 = 10000

Charles can create protein by using 10000 different sequences.

So, the probability of occurrence of each sequence is 1/10000.

Step 2

The conditional probability formula is to be used in all the 4 cases, because we know the Luis sequence and it is required to find the probability of Charles sequence is also the same.

The formula to find conditional probability is,

P(A|B) = P(A and B)/P(B)

If A and B are independent, P(A|B) = P(A).

Here, the Charles sequence and Luis sequence are independent of each other.

Step 3

(a)

It is required to match the sequence of Charles given that Luis sequence is G,H,I,J.

So, the required probability is:

P(Charles sequence is G,H,I,J | Luis sequence is G,H,I,J) = P(Charles sequence is G,H,I,J)

As expl...

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