Question

In: Statistics and Probability

A hat contains a number of cubes: 3 red, 2 white, 1 blue, and 4 black....

A hat contains a number of cubes: 3 red, 2 white, 1 blue, and 4 black.

  1. If one cube is chosen at random, what is the probability that it is:
  1. A red cube? (3 points)
  2. Not a red cube? (3 points)
  3. A cube that is white OR black? (4 points)
  4. A cube that is neither white nor black? (4 points)
  5. What do the answers to part a and part b add up to and why? (5 points)
  1. If three cubes are chosen at random, with replacement, what is the probability that:
  1. All three cubes are white? (4 points)
  2. None of the cubes are white? (4 points)
  3. At least one of the cubes is white? (4 points)
  4. The first cube is red, and the next two are black (4 points)
  1. Explain how you could simulate this experiment using a random number table. (5 points)

Solutions

Expert Solution

Total cubes = 10

a) P(red cube) = n(red cube) /10 = 3/10 = 0.3

Therefore 0.3 is the required probability here.

b) P(not a red cube) = 1 - P(red cube) = 1 - 0.3 = 0.7

Therefore 0.7 is the required probability here.

c) P(white or black) = P(white) + P(black) = 0.2 + 0.4 = 0.6

Therefore 0.6 is the required probability here.

d) P(neither black nor white) = 1 - P(white or black) = 1 - 0.6 = 0.4

Therefore 0.4 is the required probability here.

e) Sum of the probabilities in a and b are computed here as:

P(red cube) + P(not a red cube) = 0.3 + 0.7 = 1

As the we are adding complement events here, therefore the sum of probabilities here is 1.

f) As the cubes are taken out with replacement, the probability that all are white is computed here as:

= [P(white)]3

= 0.23 = 0.008

Therefore 0.008 is the required probability here.

g) P(none are white) = 0.83 = 0.512

Therefore 0.512 is the required probability here.

h) Probability that at least one of them is white

= 1 - Probability that none are white

= 1 - 0.512

= 0.488

Therefore 0.488 is the required probability here.


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