Question

We roll two fair 6-sided dice, A and B. Each one of the 36 possible outcomes is assumed to be equally likely.

a. Find the probability that dice A is larger than dice B.

b. Given that the roll resulted in a sum of 5 or less, ﬁnd the conditional probability that the two dice were equal.

c. Given that the two dice land on different numbers, ﬁnd the conditional probability that the two dice differed by 2.

Answer #1

a)there are 30 ways in which outcomes are unuequal , and out of this 30, exactly half 15 are when A is greater than B and in 15 B is greater than A

P( dice A is larger than dice B ) =15/36 =5/12

b)

number of ways sum is 5 or less =N(sum is 2)+N(sum is 3)+N(sum is 4)+N(sum is 5)=1+2+3+4=10

number of ways sum is 5 or less and both dice are equal =2 (which are (1,1),(2,2))

therefore conditional probability that the two dice were equal given sum is 5 or less =2/5

c)

number of ways two dice are different =30

number of ways dice are different and dice differed by 2 =8

therefore probability =8/30 =4/15

Consider rolling two fair six-sided dice.
a) Given that the roll resulted in sum of 8, find the
conditional probability that first die roll is 6.
b) Given that the roll resulted in sum of 4 or less, find the
conditional probability that doubles are rolled.
c) Given that the two dice land on different numbers, find the
conditional probability that at least one die is a 6.

Find the conditional probability, in a single roll of two fair
6-sided dice, that the sum is less than 6, given that the sum is
odd.

You roll two fair six-sided dice. What is the probability that
the sum of the two dice values is exactly five? Be sure to count
all possible outcomes. (Hint: The event space has 36 distinct
outcomes).

4 fair 10-sided dice are rolled.
(a)
Find the conditional probability that at least one die lands on
3 given that all 4 dice land on different numbers.
(b)
True or False: If X is the sum of the 4 numbers from
one roll, and Y is the maximum of the 4 numbers from one
roll, then X and Y are independent random
variables.

We roll three fair six-sided dice.
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(b) What is the probability that we roll a sum of at least
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(c) Now we roll three fair dice n times. How large need n be in
order to guarantee a better
than 50% chance of rolling a sum of at least 15, at least once?

5 fair 8-sided dice are rolled.
(a)
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Two fair six-sided dice are rolled once. Let (X, Y) denote the
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b) Find the probability that the face of at least one of the
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d) Given that X less than or equal to 4 find the probability
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(a)
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(b) Suppose now that each of the 7 6-sided dice are weighted the
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