Question
In: Statistics and Probability
Consider an experiment where fair die is rolled and two fair coins are flipped. Define random...
Consider an experiment where fair die is rolled and two fair coins are flipped. Define random variable X as the number shown on the die, minus the number of heads shown by the coins. Assume that all dice and coins are independent.
(a) Determine f(x), the probability mass function of X
(b) Determine F(x), the cumulative distribution function of X (write it as a function and draw its plot)
(c) Compute E[X] and V[X]
Solutions
orchestra answered 1 year agoRelated Solutions
Two fair coins and a fair die are tossed. Find the sample space of the experiment...
Two fair coins and a fair die are tossed. Find the sample space
of the
experiment (10 pts); Find the probabilities of the following
events:
A- ”the die shows 2 or 3” (5 pts);
B- ”one of the coins is head, the other - tail, and the die shows
an odd number” (5
pts).
Are the events A and B independent? (5 pts).
Give proofs.
If a fair coin is flipped twice and a standard 6 sided die is rolled once,...
If a fair coin is flipped twice and a standard 6 sided die is
rolled once, what is the likelihood of getting two 'heads' on the
coin and a '1' on the die? ​Express your answer as a percent
rounded to the tenth place
1- Two coins are flipped, followed by rolling a die as many times as the number...
1-
Two coins are flipped, followed by rolling a die as many times
as the number
of heads shown.
(a) What is the probability of getting fewer than 5 dots in
total?
(b) Given that there were exactly 3 dots in total, what is the
conditional
probability that the coins showed exactly one head?
2-
Four people are dealt 13 cards each. You (one of the players)
got one ace.
What is the probability that your partner has the other...
A fair red die and a fair green die are rolled. (a) What is the probability...
A fair red die and a fair green die are rolled.
(a) What is the probability that the sum of the numbers is
even?
(b) What is the probability that the number on the red die is
more than the number on the green die?
(c) What is the probability that the number on the red die is
twice the number on the green die?
(d) What is the probability that the number on the red die is
different from...
Two fair coins are flipped at the same time. 1) What is the probability of getting...
Two fair coins are flipped at the same time.
1) What is the probability of getting a match (same face on both
coins)? Answer for part 1 [The answer should be a number
rounded to five decimal places, don't use symbols such as
%]
2) What is the probability of getting at least two heads? Answer
for part 2 [The answer should be a number rounded to five
decimal places, don't use symbols such as %]
Consider independent trials of flipping fair coins (outcomes are heads or tails). Define the random variable...
Consider independent trials of flipping fair coins (outcomes are
heads or tails). Define the random variable T to be the first time
that two heads come up in a row (so, for the outcome HT HT HH... we
have T = 6).
(a) Compute P(T = i) for i = 1, 2, 3, 4, 5.
(b) Compute P(T = n) for n > 5.
Five fair die are rolled. What is the probability of at most two of the dice...
Five fair die are rolled. What is the probability of at most two
of the dice coming up a one or a six?
Your workings should show the use of appropriate laws and
formulae — do not provide a purely arithmetic answer.
A fair die is rolled 20 times. We define Ai as the event that number i...
A fair die is rolled 20 times. We define Ai as the event that
number i appears i
times (i = 1, ..., 6).
(a) Calculate P(Ai) for all i.
(b) Calculate P(Ai ∩ Aj) for all i ̸= j.
(c) Calculate P(A1 ∩A2 ∩A3 ∩A4 ∩A5 ∩A6). (d) Calculate P(A2 ∩A3 ∩A4
∩A5 ∩A6).
A fair die is rolled 20 times. We define Ai as the event that number i...
A fair die is rolled 20 times. We define Ai as the event that
number i appears i
times (i = 1, ..., 6).
(a) Calculate P(Ai) for all i.
(b) Calculate P(Ai ∩ Aj) for all i ̸= j.
(c) Calculate P(A1 ∩A2 ∩A3 ∩A4 ∩A5 ∩A6). (d) Calculate P(A2 ∩A3 ∩A4
∩A5 ∩A6).
Seven fair coins are flipped. The outcomes are assumed to be independent. Let X be the...
Seven fair coins are flipped. The outcomes are assumed to be
independent. Let X be the number of heads.
What is the probability that X < 3?
What is the probability that X ≥ 4?
What is the probability that 3 ≤ X < 7
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