Question

14. The table below ~hows the probabilities of all realizations
for each of the three dice. Notice that the dice are all
"loaded" meaning that they are unbalanced in some way
leading to an unequal probability of any number coming
up.

	1	2	3	4	5	6
Die 1	3.6%	30.7%	8.3%	6.1%	25.6%	25.7%
Die 2	43.0%	1.2%	17.3%	21.9%	14.5%	2.1%
Die 3	12.1%	30.2%	1.0%	18.2%	12.3%	26.2%

1. If die 3 is rolled, what is the probability of rolling a
two?
b. If one of these three dice is selected at random and
roll ed, what is the probability of rolling a two?
c. If one of these three dice is selected at random and
rolled, what number is the least likely to appear?
d. If one of these three dice is selected at random and a
two is rolled, what is the probability that die 3 was
rolled?
e. If one of these three dice is selected at random and a
four is rolled, what die is the most likely to have been
chosen?
f. If die 1 and 2 are selected and rolled, and one adds
their realizations, what is the probability that this
sum is 7?
g. If die 1 and 2 are selected and rolled, and one adds
their realizations, what is the probability that this
sum is less than S?
h. If die 1 and 2 are selected and rolled, and one adds
their realizations, what is the sum is that is more
likely to appear?
i. If die 2 and 3 are selected and rolled, and one adds
their realizations, what is the probability that this
sum is less than 11?
j. If die 1 and 3 are selected and rolled, what is the
probability that the minimum of the two realizations
is 2?
k. If die 1 and 3 are selected and rolled, what is the
probability that the minimum of the two realizations
is 5?
I. If die 1 and 3 are selected and rolled, what is the
probability that the minimum of the two realizations
is less than 4?
m. If die 2 and 3 are selected and rolled, what is the
probability that the maximum of the two realizations
is greater than 6?

Answer 1

(since there are more than 4 parts i will answer first 4)

1. If die 3 is rolled, the probability of rolling a two is the ratio of no. of times 2 shows in dice 3 and total rolls, which is the percentage given in table

P(probability of rolling a two with dice 3) = 30.2% = 0.302

b. Dice is selected at random and rolled, the probability of rolling a two = P(2)

P(2) = P(dice 1 selected and 2 comes) + P(dice 2 selected and 2 comes) + P(dice 3 selected and 2 comes)

{Probability of selecting any dice is 1/3}

P(2) = (1/3)*30.7% + (1/3)*1.2% + (1/3)*30.2%

P(2) = 0.207

c. P(x) = P(dice 1 selected and x comes) + P(dice 2 selected and x comes) + P(dice 3 selected and x comes)

P(x) = (1/3)*P(x comes in dice 1) + (1/3)*P(x comes in dice 2) + (1/3)*P(x comes in dice 3)

P(x) = (1/3)*(P(x comes in dice 1) + P(x comes in dice 2) + P(x comes in dice 3))

therefore least P(x) will be on with lowest (P(x comes in dice 1) + P(x comes in dice 2) + P(x comes in dice 3)

calculate sum of percentages of each number from 1 to 5

on comparing, we can say p(x) is least for 3

d. P(dice 3 | 2 is rolled)

= P(dice 3 selected and 2 comes) / P(2 comes)

{P(2) calculated in part b}

= ((1/3)*30.2%) / 0.207

P(dice 3 | 2 is rolled) = 0.4863

P.S. (please upvote if you find the answer satisfactory)