Question

A coin is tossed twice. Consider the following events.

A: Heads on the first toss.

B: Heads on the second toss.

C: The two tosses come out the same.

(a) Show that A, B, C are pairwise independent but not independent.

(b) Show that C is independent of A and B but not of A ∩ B.

Answer #1

If a coin is tossed twice.

(H H), (H T), (T T), (T H) = 4 outcomes

A: Heads on the first toss. (H H), (H T)

B: Heads on the second test. (H H), (T H)

C: The two tosses come out the same. (H H) (T T)

a) A, B and C are pairwise independent if

Hence A,B and C are pairwise independent.

b) C is independent of only if

hence

Thus C is not independent of

We toss a fair coin twice, define A ={the first toss is head}, B
={the second toss is tail}, and C={the two tosses have different
outcomes}. Show that events A, B, and C are pairwise independent
but not mutually independent.

Suppose you toss a fair coin three times. Which of the following
events are independent? Give mathematical justification for your
answer.
A=
{“heads on first toss”}; B=
{“an odd number of heads”}.
A=
{“no tails in the first two tosses”}; B=
{“no heads in the second and third
toss”}.

A fair six-sided die is tossed twice. Consider the following
events:
1. A : First toss yields an even number.
2. B : Second toss yields an odd number.
3. C : Sum of two outcomes is even.
Find P(A), P(B), P(C), P(B ∩ C), P(C|B), P(C|A ∩ B) and
P(B|C).

I toss a coin two times. X1 is the number of heads on the first
toss. X2 is the number of heads on the second toss.
Find the mean of X1.
Find the variance of X1.
Find the mean of X1 + X2. (This is the number of heads in 2
tosses.)
Find the variance of X1 + X2.
If you tossed 10 coins, how many heads would you expect? What is
the variance of the number of heads?

A penny which is unbalanced so that the probability of heads is
0.4 is tossed twice. Let
Z be the number of heads obtained in the
first toss. Let W be the total number of heads
obtained in the two tosses of the coin.
a)Calculate the correlation
coefficient between Z and W
b)Show numerically that Cov(Z, W) =
VarZ
c)Show without using numbers that
Cov(Z, W) = VarZ

(a) A fair coin is tossed five times. Let E be the event that an
odd number of tails occurs, and let F be the event that the first
toss is tails. Are E and F independent?
(b) A fair coin is tossed twice. Let E be the event that the
first toss is heads, let F be the event that the second toss is
tails, and let G be the event that the tosses result in exactly one
heads...

Suppose we toss a fair coin three times. Consider the events
A: we toss three heads, B: we toss at least one
head, and C: we toss at least two tails.
P(A) = 12.5
P(B) = .875
P(C) = .50
What is P(A ∩ B), P(A ∩ C) and P(B ∩ C)?
If you can show steps, that'd be great. I'm not fully sure what
the difference between ∩ and ∪ is (sorry I can't make the ∪
bigger).

A balanced coin is tossed 3 times, and among the 3 coin tosses,
X heads show. Then the same balanced coin is tossed X additional
times, and among these X coin tosses, Y heads show.
a. Find the distribution for Y .
b. Find the expected value of Y .
c. Find the variance of Y .
d. Find the standard deviation of Y

A balanced coin is tossed 3 times, and among the 3 coin tosses,
X heads show. Then the same balanced coin is tossed X additional
times, and among these X coin tosses, Y heads show.
a. Find the distribution for Y .
b. Find the expected value of Y .
c. Find the variance of Y .
d. Find the standard deviation of Y

Alan tosses a coin 20 times. Bob pays Alan $1 if the first toss
falls heads, $2 if the first toss falls tails and the second heads,
$4 if the first two tosses both fall tails and the third heads, $8
if the first three tosses fall tails and the fourth heads, etc. If
the game is to be fair, how much should Alan pay Bob for the right
to play the game?

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 9 minutes ago

asked 11 minutes ago

asked 17 minutes ago

asked 34 minutes ago

asked 35 minutes ago

asked 40 minutes ago

asked 43 minutes ago

asked 45 minutes ago

asked 49 minutes ago

asked 1 hour ago

asked 1 hour ago