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The Unfair Die

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You are given a loaded six-sided die. Such a die can produce any integer from 1 to 6. The die rolls a 4 three times as often as any other number.
What is the probability of getting a 4?
You now roll the die 5 times as an experiment. What are the chances of the following scenarios? Instead of solving for event and sample spaces, use the probabilities you found in part I. Show all of your work.
Die lands on 5 unique values?
Die lands on the same value 5 times?
Die lands on 4 exactly 3 times?

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Step-by-step computations and explanations are given in the solutions.

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I.
The five numbers 1, 2, 3, 5 and 6 appear equally often, i.e. they have equal probabilities of being rolled.
Suppose the die is rolled once.
Let x = probability of rolling a 1
= probability of rolling a 2
= probability of rolling a 3
= probability of rolling a 5
= probability of rolling a 6

Since the probability of rolling a 4 is three times as often as any other number, the probability of rolling a 4 is 3x.

5x + 3x = 1
8x = 1
x = 1/8
3x = 3/8
The probability of rolling a 4 is 3/8.

II.
A.
Case 1—the number 4 is not one of the 5 numbers ...

Solution provided by:
J Pye, MSc
Education
  • MSc, California State Polytechnic University, Pomona
  • MBA, University of California, Riverside
  • BSc, California State Polytechnic University, Pomona
  • BSc, California State Polytechnic University, Pomona
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