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1. Answer the following.

1.1 Probability of < 2 or 5 successes given a Binominal distribution with 10 trials and probability of success .2 [Hint: binCDF]

1.2 Probability of < 2 and 5 successes given a Binominal distribution with 10 trials and probability of success .2

1.3 If an urn contains 5 white, 3 blue, and 2 red jelly beans, identical in all but color, determine how many ways they can be uniquely arranged without replacement [Hint: Partitioning]

1.4 Probability of drawing 5 non-face cards --- cards other than Ace, King, Queen, and Jack --- of the same suit in a five card hand dealt from a standard intact 52 card deck [Hint: Dependent events]

1.5 Probability of winning the mega lottery given the purchase of a ticket and given the selection of a five digit number which has no repeated digits [Hint: Key words: No repeated digits]

9. If errors in unedited manuscripts are Poisson distributed with 3 errors in 5 pages and if each manuscript entails 4 pages, answer the following. [Hint: Poisson distribution]

9.1 Probability of 1 or 3 errors in a manuscript

9.2 Probability of 4 or more errors in a manuscript

9.3 Probability of exactly 3.5 errors in a manuscript

9.4 Given three individual manuscripts, the probability that one of them will contain 4 or more errors and the other two will contain less than 4 errors

9.5 Given three combined manuscripts, the probability that the combination will contain more than 4 errors.

3. In serially drawing three cards without replacement from a standard 52 card deck, determine the following probabilities. [Hint: Note whether order is required and note that all are dependent events]

3.1 First three cards are queens

3.2 First three cards are red

3.3 Third card is red given that the first two cards were black

3.4 First three cards are a king, queen, and ace in that order

3.5 First three cards are a king, queen, and ace in any order

3.6 First three cards are the same suit

2. Answer the following assuming fair dice; note that the event space with two die includes 36 distinct pair of outcomes.

2.1 Given the sum of the face values, determine whether a sum of six, seven, or eight is more probable in rolling two dice [Hint: Determine the total space and then within that space the respective combinations --- order important --- that yield the given sum]

2.2 Determine the expected return in betting $10 and potentially winning $50 if winning entails getting the same face value on two dice rolled, namely, winning $50 and retaining $10 on a winning bet and losing the $10 on a losing bet [Hint: Expected value = win amount * win probability loss amount * loss probability]

2.3 Determine the probability of rolling snake eyes (a pair of twos) a fifth time after having done that on the previous four consecutive rolls of the dice [Hint: Independent events]

2.4 Determine the probability of rolling snake eyes at least once on five consecutive rolls of the dice. [Hint: Binominal distribution]

4. If three (straight) couples find six adjacent empty seats at the theater, determine the number of ways they can be seated. [Hint: Factorial; treat the special circumstances first]

4.1 No order whatsoever

4.2 The men sit adjacent to each other and the women sit adjacent to each other

4.3 Each couple sits together

4.4 Men and women sit in alternate seats

4.5 The men sit adjacent to each other

4.6 A couple sits together

4.7 A couple sits together in the first two or the last two seats

4.8 Men and women sit in pairs but each person is not next to his/her threater date.

6. Assuming the consistent historical failure rate in testing finished units is 1 in 200 and assuming 25 units are randomly selected and tested from each 1000 unit production run, answer the following. [Hint: Binominal distribution]

6.1 Expected number of failures

6.2 Probability exactly one fails

6.3 Probability at least one fails

6.4 Probability at most one fails

6.5 Probability that, if the given test is repeated 4 times, it will result in 2 of the 4 repetitions having exactly one failure each

11. Given the objective of getting exactly two heads in each repetition of four flips of three coins, determine the probability distribution and its expected value and variance.