A bag contains 9 red marbles, 5 white marbles, and 6 blue marbles. You draw 4 marbles out at random, without replacement. Find the following probabilities and round to 4 decimal places.

a) 0.0260

b) 0.0681

Explanation:

a)

The bag contains:

r = 9 (number of red marbles)

w = 5 (number of white marbles)

b = 6 (number of blue marbles)

So, the total number of marbles in the bag at the beginning is:

At the 1st attempt, the probability of choosing a red marble is:

(1)

At the 2nd attempt, the 1st red marble is not placed back, so now the number of marbles is (n-1), while the number of red marbles left is (r-1). So the probability of choosing another red marble at the 2nd attempt is

(2)

With a similar argument, the probabilities of selecting a red marble in the 3rd and 4th attempt are

(3)

(4)

Therefore, the probability of drawing 4 red marbles in the first 4 attempts without replacing is:

b)

At the 1st draw, the probability that the marble is not red is:

At the 2nd draw, there are 9 red marbles left and 19 total marble left. So, the probability of NOT drawing a red marble is:

At the 3rd draw, there are 9 red marbles left and 18 total marbles left. So, the probability of NOT drawing a red marble is:

Finally, with a similar argument the probability of NOT drawing a red marble at the 4th and last attempt is:

So, the total probability of drawing 4 non-red marbles in the first 4 attempts is:

answer; is usually (less than) the socially optimal level of consumption;

a manufacturer of printed circuit boards uses exponential smoothing with trend to forecast monthly demand of its product. at the end of december, the company wishes to forecast sales for january. the estimate of trend through november has been 200 additional boards sold per month. average sales have been around 1000 units per month. the demand for december was 1100 units. the company uses α = 0.20 and β = 0.10. make a forecast including trend for the month of january.