Full credit will only be given to correct answers with a clear explanation of how they are obtained. Use additional paper as necessary.

   Imagine a tetrahedral die with four faces labeled "A", "C", "G", and "T" occurring with probabilities P(A)= 0.3,  P(C) = 0.2, P(G) = 0.25, and 

1. The nucleotide base pair C and G can form 3 hydrogen bonds between them, while A and T can form only 2. The bonding between C and G is therefore stronger than that between A and T. For this reason, C and G are called "strong" bases, while A and T "weak" bases. According to the chemical structure of the bases, A and G are classified as "purines" while C and T "pyrimidines". Write down the probabilities of the events S = {C, G}, W = {A, T}, R = {A, G}, Y = {C, T}. (Hint: Recall the Axiom of Finite Additivity).
   
    A. P(S) = 0.05, P(W) = 0.075, P(R) = 0.075, P(Y) = 0.05
    B. P(S) = 0.45, P(W) = 0.55, P(R) = 0.55, P(Y) = 0.45
    C. P(S) = 1.35, P(W) = 1.10, P(R) = 1.35, P(Y) = 1.10
    D. None of the above.

 

2. When this tetrahedral die is rolled two times, what is the probability that the outcomes of the two rolls are both strong bases? Both weak bases? One strong base and one weak base? Both strong bases or both purines?  (Hint: At least two rules of probability will be used in this problem. The Axiom of Finite Additivity and the rule for taking the intersection of independent events: Multiplication Rule. Note that "and" corresponds to "intersection" and "or" corresponds to "union" ).

   Both strong bases:
    A. 0.1000
    B. 0.9000
    C. 0.1505
    D. 0.2025

Both weak bases:
 A. 0.1500
 B. 0.0908
 C. 0.3025
 D. 0.5000

One strong and one weak base?
 A. 0.4950
 B. 0.2475
 C. 0.5050
 D. 0.1200

Both strong bases or both purines:
Hint: Use the following Theorem: If P is a probability function and A and B are any sets in S, then

P(A or B) = P(A) + P(B) - P(A and B)
 A. 0.5050
 B. 0.4425
 C. 0.0000
 D. 0.5675
3. Given that the tetrahedral die is rolled two times , and both rolls result in a strong base. What is the probability that at least one of these rolls is a purine? (Hint)
    A. 0.7525
    B. 0.8525
    C. 0.8025
    D. 0.2025

 

4. When this tetrahedral die is rolled three times, what is the probability that the sequence of bases generated is not a stop codon?  (Hint: Recall that if P is a probability function and A is any set, then P( A^c ) = 1 - P(A). So, find P(A), where A=getting a stop codon, and then subtract this probability from 1.).

    A. 0.5000
    B. 0.9400
    C. 0.0600
    D. 0.6000

 

5. When this die is rolled four times, what is the probability that the sequence of bases generated is a palindrome? (A palindrome is a sequence of bases which is the same as its complementary sequence read in the reverse direction. AATT, GATC are examples. Biologically, you can think of palindrome as a fragment of DNA which is identical on both strands).

    A. 0.3275
    B. 0.0175
    C. 0.0625
    D. 0.0463

6. Review the Law of Total Probability and Bayes Theorem for next class.