A fair 6-sided die is rolled repeatedly. (a) Find the expected
number of rolls needed to get a 1 followed right away by a 2. Hint:
Start by conditioning on whether or not the first roll is a 1. (b)
Find the expected number of rolls needed to get two consecutive
1’s. (c) Let an be the expected number of rolls needed to get the
same value n times in a row (i.e., to obtain a streak of n
consecutive...
Alice rolls a pair of fair six-sided dice until a sum of 6
appears for the first time. Independently, Bob rolls two fair
six-sided dice until he rolls a sum 7 for the first time. Find the
probability that the number of times Alice rolls her dice is equal
to or within one of the number of times Bob rolls his dice.
Suppose a 6-sided die and a 7-sided die are rolled. What is the
probability of getting sum less than or equal to 5 for the first
time on the 4th roll? Show your work to receive credit.
Suppose you roll a fair 15 sided die. The numbers 1-15 appear
once on different sides. (Imagine a regular die with 12 sides
instead of 6.) Answers may be left in formula form.
(a) What is the probability of rolling a 7?
(b) What is the probability of rolling an odd number and a
number greater than 8?
(c) What is the probability of rolling an even number or a
number greater than 9?
(d) Suppose you roll the die,...
1. In 90 rolls of a six-sided die, the outcome of 1 appears 16
times. State whether the difference between what occurred and what
you would have expected by chance is statistically significant. Is
the difference between what occurred and what is expected by chance
statistically significant?
No
Yes
2. What is statistical inference? Why is it important?
A.Statistical inference is the process of determining if the
difference between what is observed and what is expected is too
great to...
Problem 28. A fair six-sided die is rolled repeatedly and the
rolls are recorded. When two consecutive rolls are identical, the
process is ended. Let S denote the sum of all the rolls made. Is S
more likely to be even, odd or just as likely even as odd?
Compute ?(?) for the following random variable ?:
?=Number of tosses until all 10 numbers are seen (including the
last toss) by tossing a fair 10-sided die.
To answer this, we will use induction and follow the steps
below:
Let ?(?) be the expected number of additional tosses until all
10 numbers are seen (including the last toss)
given ? distinct numbers have already been
seen.
1. Find ?(10).
?(10)=?
2.
Write down a relation between ?(?) and ?(?+1). Answer...