MCB 432 Assignment 3 Basic Probability Due Feb 13 2014 Name key 1 pt for name 40 pt total Report answers as a reduced fraction or decimal fraction e g 17 19 or 0 8947 not 2329 2603 Unless I say otherwise you may round to 3 significant figures No partial credit without comprehensible work 1 Assume one or more fair dice with sides numbered 1 8 each of the 8 numbers is equally likely 1a When a die is thrown what is the probability of a 5 1 pt 1 8 0 125 1b A die is thrown and a 1 turns up The next time it is thrown what is the probability of a 5 1 pt 1 8 0 125 the events are independent the probability does not depend on the history 1c When a die is thrown what is the probability that the result will be either a 1 or a 5 2 pt P 1 or 5 P 1 P 5 1 8 1 8 1 4 0 25 alternative solutions is sum 1d When two dice are thrown what is the probability that they both turn up 3 2 pt P 3 3 P 3 P 3 1 8 1 8 1 64 0 0156 1 56 10 2 2 conditions is product 1e When a die is thrown twice what is the probability that the first throw is a 3 and the second throw is a 4 2 pt P 3 4 P 3 P 4 1 8 1 8 1 64 0 0156 1 56 10 2 2 conditions is product 1f When a die is thrown twice what is the probability that the results include a 3 and a 4 2 pt P 3 4 or 4 3 P 3 4 P 4 3 2 1 8 1 8 2 64 1 32 0 0313 3 13 10 2 1g When two dice are thrown what is the probability that the total of the numbers appearing is 5 2 pt P 5 total P 1 4 P 2 3 P 3 2 P 4 1 4 1 8 1 8 1 16 0 0625 6 25 10 2 2 n n i m i m f i is a summation It is defined as f i f m f m 1 f m 2 f n where f i is a specified function of i It is commonly used in situations where there are multiple ways of satisfying a condition such as what is the probability of 5 or more heads in 10 tosses of a fair coin or what is the probability of 3 or fewer restriction sites in a genome 4 2a What is the exact value of i i 1 2 pt 1 2 3 4 10 4 2b What is the exact value of i i 0 2 pt 0 1 2 3 4 10 2 2c What is the exact value of i 2 i 0 2 2 2 2 pt 0 1 2 0 1 4 5 17 2d What is the exact value of 17 17 evaluated at i 7 or any other value is 17 done 11 times i 7 2 pt 17 17 17 17 17 17 17 17 17 17 17 11 17 187 5 2e What is the exact value of 3i i 3 3 4 5 2 pt 3 3 3 27 81 243 351 Page 2 Assignment 3 Name 3 is the factorial operator which we will use in defining the probabilities specific outcomes in the Binomial and Poisson distributions For n 0 by definition n n n 1 3a The above definition lacks the value for n 0 which is essential to getting the series started What is the value of 0 1 pt 1 3b What is the exact value of 6 2 pt 6 5 4 3 2 1 720 10001 3c What is the exact value of 9998 2 pt 10001 10000 9999 9998 9998 10001 10000 9999 999 999 990 000 n n i m i m 4 f i is the product operator and is defined by f i f m f m 1 f m 2 f m 3 f n where f i is a specified function of i It is analogous to the summation operator but the values of n the function are multiplied rather than added For example for n 0 n i This is common i 1 is likelihood calculations where many separate conditions must be simultaneously met 4 4a What is the exact value of i i 1 2 pt 1 2 3 4 24 which is 4 4 4b What is the exact value of i i 0 2 pt 0 0 the zero factor at i 0 makes everything else moot 13 4c What is the exact value of 3 i 10 4 2 pt 3 3 3 3 3 81 3 evaluated at i 10 or any other value is 3 done 4 times 2 4d What is the exact value of 4i i 0 0 1 2 3 2 pt 4 4 4 4 64 or 40 41 42 1 4 16 64 5a Assume a random DNA sequence with a base composition of A C G T 0 30 0 20 0 20 0 30 i e 40 G C What is the probability that any particular hexanucleotide in the molecule has the sequence GAATTC an EcoRI restriction site 2 pt P GAATTC P G P A P A P T P T P C 0 2 0 3 0 3 0 3 0 3 0 2 3 24 10 4 5b What is the probability that any particular hexanucleotide in the molecule has the sequence GGATCC a BamHI restriction site in A T rich DNA a G C rich site is rarer 2 pt P GGATCC P G P G P A P T P C P C 0 2 0 2 0 3 0 3 0 2 0 2 1 44 10 4