# Calculate the final probabilities of the four possible junk–keyword outcome combinations, i.e. the probability of an email being ”junk” and containing the ”offer” word, ”non-junk” and containing the ”offer” word etc.

STATISTICS
Paper, Order, or Assignment Requirements

Use correct probability notation in your responses to all questions. Marks are awarded for presentation including correct notation.
Correct probability notation implies statements such as:

P(X<5) = 0.1, P(5<X<10) = 0.65 etc,

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Use the probability notation and do not substitute it with words.

Questions

1 Junk email
A systems administrator did a quick analysis and found that 35% of the emails in his inbox can be classified as junk. In addition, he noted that 80% of junk emails contain the word offer in their subject line, whereas, only 15% of the non-junk emails have a similar subject line.
(a) Calculate the final probabilities of the four possible junk–keyword outcome combinations, i.e. the probability of an email being ”junk” and containing the ”offer” word, ”non-junk” and containing the ”offer” word etc. You might find it useful to create a tree diagram illustrating the above information but it is not necessary to include it in your submitted manuscript.
(b) Show that the overall probability of an email containing the keyword ”offer”, regardless of its ”junk” or ”non-junk” status, is 0.345.
(c) Calculate the conditional probability that an email is junk given that it contains the keyword ”offer”.

2 Cell cultures
Many lab-based experiments rely on cultured cells. Culturing cells can be a difficult process, and many culturing attemps fail. In one particular lab, numerous cultures are begun, and only those cultures that thrive are used in subsequent experiments.
Suppose that the probability that any one culture will succeed is 0.2, the success or failure of any given culture does not depend on or influence the success or failure of the other cultures, and 10 cultures are attempted.
(a) Let X be the number of successful cultures out of the 10 attempted cultures. Explain why X is a binomial random variable.
Using Rcmdr, plot:
(b) the probability function for this example
(c) the cumulative probability function (distribution function) for this example.
Using Rcmdr, calculate:
(d) At most 3 (i.e. less or equal to 3) cultures are successful.
(e) At least 3 (i.e. greater or equal to 3) cultures are successful
(f) Exactly 3 cultures are successful.

3 Biochemical reaction
In a stable biochemical system a reaction occurs, on average, 3.5 times per minute. Assume that the number of reaction occurrences per minute is a random variable (say X) that follows a Poisson distribution. Using Rcmdr calculate the probability of:
(a) no reaction occurring during a minute.
(b) 1 to 5 reactions occurring during a minute.
(c) more than 5 reactions occurring during a minute.