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.
Consider the setting where you first roll a fair 6-sided die, and then you flip a fair coin the number of times shown by the die. Let D refer to the outcome of the die roll (i.e., number of coin flips) and let H refer to the number of heads observed after D coin flips. (a) Suppose the outcome of rolling the fair 6-sided die is d. Determine E[H|d] and Var(H|d). (b) Determine E[H] and Var(H).
suppose you only have one fair 6-sided die. We will say that a success is if you roll a 5 or a 6. You roll the die over and over until you roll two successes in a row. What is the the expected number of times you must roll before you stop?
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].
We flip a coin. If it is heads we roll a four sided die with sides numbered from 1 to 4. If it is tails, we roll a six sided die with sides numbered from 1 to 6. We let X be the number rolled. (a) What is the expectation of X? (b) What is the variance of X? (c) What is the standard deviation of X? We draw cards one by one and with replacement from a standard deck...
If I flip a coin and roll a six sided die simultaneously, the SAMPLE SPACE of this experiment holds ________ possible unique outcomes. A.36 B.24 C.12 D.8
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 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 I asked you to roll a fair six-sided die 6 times. You have already rolled the die for 5 times and six has not appeared ones. Assuming die rolls are independent, what is the probability that you would get a six in the next roll? 1/6 1/2 5/6 0 1
Draw a tree diagram displaying all possible outcomes given a roll of a six sided die, a coin flip, then a second coin flip. Examples of possible outcomes are 5H H meaning a roll of a , then a Heads, and a Heads TH meaning a roll of a 1, then a Tails, then a Heads
3) We roll 2 fair dice. a) Find the probabilities of getting each possible sum (i.e. find Pr(2), Pr(3), . Pr(12) ) b) Find the probability of getting a sum of 3 or 4 (i.e.find Pr(3 or 4)) c) Find the probability we roll doubles (both dice show the same value). d) Find the probability that we roll a sum of 8 or doubles (both dice show the same value). e) Is it more likely that we get a sum...