Question

Probability. Read prompt and answer 3 questions.

To play a game, you have a bag containing:

•  30 fair six-sided dice, with faces {1,2,3,4,5,6}.

• 17 fair eight-sided dice (faces {1,2,3,4,5,6,7,8})

• and 3 fair twenty-sided dice (faces {1,2,3,...,19,20})

Call these 3 classes of die "Six", "Eight" and "Twenty" (or S, E, and T, for short).

Please show your work and answer the following questions:

Part A: You roll your die one time. What is the probability of the event ?7R7, that you roll a 7? Include at least 4 decimal places in your answer.

Part B: Suppose you roll a 7. What is the probability that the die you chose from the bag is an Eight-sided die?

Part C: Are the events ?7R7 and ?E independent? Justify your answer using math.

Answer 1

30 fair six-sided dice, with faces {1,2,3,4,5,6}

17 fair eight-sided dice (faces {1,2,3,4,5,6,7,8})

3 fair twenty-sided dice (faces {1,2,3,...,19,20})

total number of die = 30 + 17 + 3 = 50

P[ fair six-sided dice ] = P[ S ] = number of fair six-sided dice / total number of dice

P[ S ] = 30/50 = 0.6

P[ fair eight-sided dice ] = P[ E ] = number of fair eight-sided / total number of dice

P[ E ] = 17/50 = 0.34

P[ fair twenty-sided dice ] = P[ T ] = number of fair twenty-sided / total number of dice

P[ T ] = 3/50 = 0.06

A)

P[ getting roll of 7 on fair six-sided die ] = P[ R7 | S ] = 0 ( 7 can not appear )

P[ getting roll of 7 on fair eight-sided die ] = P[ R7 | E ] = 1/8 ( all outcome are equally probable )

P[ getting roll of 7 on fair twenty-sided die ] = P[ R7 | T ] = 1/20 ( all outcome are equally probable )

What is the probability of the event R7, that you roll a 7?

P[ R7 ] = P[ R7 | S ]*P[S] + P[ R7 | E ]*P[E] + P[ R7 | T ]*P[T]

P[ R7 ] = 0*0.6 + (1/8)*0.34 + (1/20)*0.06

P[ R7 ] = 0 + 0.0425 + 0.003

P[ R7 ] = 0.0455

Probability of the event R7, that you roll a 7 = 0.0455

B)

Suppose you roll a 7. What is the probability that the die you chose from the bag is an Eight-sided die?

P[ E | R7 ] = P[ R7 | E ]*P[E] / P[ R7 ]

P[ E | R7 ] = (1/8)*0.34 / 0.045

P[ E | R7 ] = 0.0425/0.0455

P[ E | R7 ] = 0.9341

C)

Are the events ?7R7 and ?E independent?

For the events to be independent , P[ A | B ] = P[ A ]

Here, P[ E | R7 ] ( 0.9341 ) != P[ E ] ( 0.34 )

Hence, the events R7 and E are not independent.