1. Suppose we have a fair die that has twelve (12) sides. That is, if we...
1. Suppose we have a fair die that has twelve (12) sides. That is, if we roll it, each of the first 12 positive integers are equally likely to be the result of the roll. (a) If we roll the die, what is the probability the result is prime? As a reminder, one is not a prime number. (b) Suppose we roll this die 1000 times. What is the probability we get a prime number exactly 200 times? (c) Suppose we roll this die 1000 times. What is the probability we get a prime number at least 200 times? (d) What is the expected value of a single roll of this die?
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1. Suppose we have a fair die that has twelve (12) sides. That
is, if we...
Suppose you are rolling a fair four-sided die and a fair six-sided die and you are...
Suppose you are rolling a fair four-sided die and a fair six-sided die and you are counting the number of ones that come up. What is the probability that both die roll ones? What is the probability that exactly one die rolls a one? What is the probability that neither die rolls a one? What is the expected number of ones? If you did this 1000 times, approximately how many times would you expect that exactly one die would roll...
We roll a fair 8-sided die five times. (A fair 8-sided die is equally likely to...
We roll a fair 8-sided die five times. (A fair 8-sided die is equally likely to be 1, 2, 3, 4, 5, 6, 7, or 8.) (a) What is the probability that at least one of the rolls is a 3? (b) Let X be the number of different values rolled. For example, if the five rolls are 2, 3, 8, 8, 7, then X = 4 (since four different values were rolled: 2,3,7,8). Find E[X].
1) Suppose we have a fair 6 sided die and a coin. a) If we roll...
1) Suppose we have a fair 6 sided die and a coin. a) If we roll the die 4 times, the total number of possible outcomes is? b) If we roll the die 2 times then flip the coin 3 times, the total number of possible outcomes is? Show your calculations.
Example 5.5. We roll a fair die then toss a coin the number of times shown...
Example 5.5. We roll a fair die then toss a coin the number of times shown on the die. What is the probability of the event A that all coin tosses result in heads? One could use the state space Ω = {(1, H), (1, T), (2, H, H), (2, T, T), (2, T, H), (2, H, T), . . . }. However, the outcomes are then not all equally likely. Instead, we continue the state space is Ω {1,...
If we roll a red 6-sided die and a green 6-sided die (both are fair dice...
If we roll a red 6-sided die and a green 6-sided die (both are fair dice with the numbers 1-6 equally likely to be rolled), what is the probability that we get (i) A 5 on the green die AND a 3 on the red die? (ii) A 5 on the green die OR a 3 on the red die? (iii) A 5 on the green die GIVEN we rolled a 3 on the red die?
Suppose that a fair die is rolled n times. We say that there is an increase...
Suppose that a fair die is rolled n times. We say that there is an increase at the i’th place if result on the i + 1’st roll is greater than the result on the i’th roll. Let X be a random variable representing the number of increases. Find E[X].
Suppose that a fair die is rolled n times. We say that there is an increase at the i’th place if ...
Suppose that a fair die is rolled n times. We say that there is an increase at the i’th place if result on the i + 1’st roll is greater than the result on the i’th roll. Let X be a random variable representing the number of increases. Find E[X].
Suppose that we roll a fair die that has two faces numbered 1, two faces numbered...
Suppose that we roll a fair die that has two faces numbered 1, two faces numbered 2, and two faces numbered 3. Then we toss a fair coin the number of times indicated by the number on the die and count the number of heads. How much information is obtained (on the average) by this procedure?
-3 points Suppose we have a fair coin that is equally likely to come up heads...
-3 points Suppose we have a fair coin that is equally likely to come up heads or tails (a) The coin is flipped 3 times. How many ways can we get exactly 1 head? (b) The coin is flipped 5 times. How many ways can we get exactly 2 tails? (c) The coin is flipped 4 times. How many ways can we get at least 3 tails?
A statistician has a 6-sided die which she does not believe is a fair die. On...
A statistician has a 6-sided die which she does not believe is a fair die. On any given roll, she suspects that the numbers 1 to 5 are all equally likely to occur, but that the number 6 is three times as likely to occur as any of the other numbers. To investigate further, she performs a study by rolling the die 100 times and recording the number of times each number occurred. The data are described below. 1 Number...
We roll a fair 8-sided die five times. (A fair 8-sided die is equally likely to be 1, 2, 3, 4, 5, 6, 7, or 8.) (a) What is the probability that at least one of the rolls is a 3? (b) Let X be the number of different values rolled. For example, if the five rolls are 2, 3, 8, 8, 7, then X = 4 (since four different values were rolled: 2,3,7,8). Find E[X].
Example 5.5. We roll a fair die then toss a coin the number of times shown on the die. What is the probability of the event A that all coin tosses result in heads? One could use the state space Ω = {(1, H), (1, T), (2, H, H), (2, T, T), (2, T, H), (2, H, T), . . . }. However, the outcomes are then not all equally likely. Instead, we continue the state space is Ω {1,...
-3 points Suppose we have a fair coin that is equally likely to come up heads or tails (a) The coin is flipped 3 times. How many ways can we get exactly 1 head? (b) The coin is flipped 5 times. How many ways can we get exactly 2 tails? (c) The coin is flipped 4 times. How many ways can we get at least 3 tails?
A statistician has a 6-sided die which she does not believe is a fair die. On any given roll, she suspects that the numbers 1 to 5 are all equally likely to occur, but that the number 6 is three times as likely to occur as any of the other numbers. To investigate further, she performs a study by rolling the die 100 times and recording the number of times each number occurred. The data are described below. 1 Number...
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