A coin that lands on heads with a probability of p is tossed
multiple times. Each toss is independent. X is the number of heads
in the first m tosses and Y is the number of heads in the first n
tosses. m and n are fixed integers where 0 < m < n. Find the
joint distribution of X and Y.
Two coins are tossed at the same time. Let random variable be
the number of heads showing.
a) Construct a probability distribution for
b) Find the expected value of the number of heads.
A jar contains 100 coins. One of the coins is a trick coin
with heads on both sides. The other 99 coins are ordinary coins
with heads probability .5. A coin is selected at random. (a) What
is the probability the coin selected is the trick coin? (b) The
coin selected is tossed 7 times, and it happens to land heads each
of those times. Now what is the probability it is the trick
coin?
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.
A coin is tossed with P(heads) = p.
a) What is the expected number of tosses required to get n
heads?
b) Determine the variance of the number of tosses needed to get
the first head.
c) Determine the variance of the number of tosses needed to get
n heads.
A coin is tossed with P(heads) = p.
a) What is the expected number of tosses required to get n
heads?
b) Determine the variance of the number of tosses needed to get
the first head.
c) Determine the variance of the number of tosses needed to get
n heads.
Five coins were simultaneously
tossed 1000 times and at each toss, the number of heads was
observed. The number of tosses during which 0, 1, 2, 4, and 5 heads
were obtained are shown in the table below. Convert the given
frequency distribution to probability distribution find the
expected value.
Number of heads per toss:
0
1
2
3
4 5
Number of tosses
: 38
144 342
287
164
25
Three fair coins are tossed simultaneously. We define the
events
A: get at least two heads
B: get two tails
C: get at most two heads
i) Write
down the sample space S.
ii) Calculate
P(A), P(B), P(C).
iii) Calculate P(A ∩
B), P(A ∩ C), P(B ∩
C).
iv) Calculate P(A Ս B),
P(A Ս C), P(B Ս
C).
v) Which of the
following pairs of events are independent and why?
A and B
A and C
vi) Which of the
following pairs of events...