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1)
A study into the survival rates of Australian Businesses found that of the businesses operating in 2007, 95.8% were classified as small businesses (i.e. they had fewer than 20 employees). If a business is not classified as a small business, the probability that it is still operating after four years is 75.7%. If a business is classified as a small business, then the probability of its still operating after four years is 59.7%.
a. Using letters of the alphabet and appropriate probability notation, define the two simple events described in this problem and their complements (4 definitions altogether).
b. Draw a probability tree to represent the information given in the question using the letters that you used to define the simple events in part a.
Note: Marks will be deducted if the probabilities do not appear along the branches of the probability tree.
c. Determine the probability that a randomly chosen business has fewer than 20 employees and will still be operating after four years.
d. Determine the probability that a randomly chosen business will no longer be operating after four years.
e. Given that a business that had been operating more than four years ago is no longer operating, what is the probability that it had been a small business?
2)
Government research has found that 78% of people aged 15 to 19 who live in a major city are attending an educational institution. The research also found that only 40% of people aged 15 to 19 who live in very remote parts of the country are attending an educational institution.
A sample of 25 people aged 15 to 19 living in very remote areas of the country is selected.
a. Identify the type of distribution being described by the random variable in the sentence above and write down the value(s) of its parameter(s).
b. Calculate the probability that less than half of these 25 people are attending an educational institution.
Another sample of one hundred 15- to 19-year-olds living in a major city is selected. The number attending an educational institution is counted.
c. Calculate the probability that more than three quarters of these 15- to 19-year-olds are attending an educational institution.
d. Calculate the probability that more than 25 but fewer than 75 of these 15- to 19year-olds are attending an educational institution.