Question

1) A certain disease has an incidence rate of 0.5%. If the false negative rate is 6% and the false positive rate is 4%, compute the probability that a person who tests positive actually has the disease. 2) Karla has 10 shirts, 6 shorts, and 3 pairs of sandals. How many different combinations of outfits could she make? 3) You get to choose your new 4 digit pin number for your debit card. If you are allowed to repeat the numbers, how many different pin numbers from which do you have to choose? 4) If 380 people ran in the Savannah Bridge Run, how many ways can first, second, and third place be awarded? 5) There are 210 employees that work the assembly line at the local automotive plant. The boss has to choose a group 5 to go to training. How many different ways can he choose the group?

Answer 1

1) The probability that a person who tests positive actually has the disease is 0.1056 or 10.56%

2) 180 combinations

3) 5040 pin numbers can be chosen

4) 54439560 ways

5) 389283935040 ways

Explanation:

1) From Bayes' theorem, we have;

`P(A\mid B) = \frac{P(A)P(B\mid A)}{P(A)P(B\mid A)+P\left (\overline{A} \right )P\left (B\mid \overline{A} \right )}`

Where:

`P(A\mid B)` = Percentage of the population that tests positive for the disease that actually have the disease

P(A) = Incidence rate = 0.5% = 0.005

`P(B\mid A)` = Percentage of the population that have the disease and will tests positive = 1 - 0.06 = 0.94

`P\left (\overline{A} \right )` = P(no disease) = 1 - 0.005 = 0.995

`P\left (B\mid \overline{A} \right )}` = P(positive
`\mid` no disease) = False positive = 0.04

From which we have;

`P(A\mid B) = (0.005 * 0.94)/(0.005 * 0.94 +0.995 *0.04) = 0.1056`

Therefore, the probability that a person who tests positive actually has the disease = 0.1056 or 10.56%

2) The parameters given are;

The number of shirts that Karla has = 10 shirts

The number of shorts that Karla has = 6 shorts

The number of pairs of sandals that Karla has = 3 pairs

Therefore;

The number of ways Karla can choose her shirts = 10 ways

The number of ways Karla can choose her shorts = 6 ways

The number of ways Karla can choose her pair of sandals = 3 ways

The total combination of outfits Karla can make = 10 × 6 × 3 = 180 combinations

3) We have;

The number of permutations of 10 numbers taking 4 at a time is given as follows;

n!/(n-r)! = 10!/(10-4)! = 10!/6! = 5040 pin numbers can be chosen

4) The number of ways the first second and the third can be awarded is 380×379×378 = 54439560 ways

5) The number of ways is 210×209×208×207×206 = 389283935040 ways.