Question

In: Statistics and Probability

Suppose a fair coin is tossed 3 times. Let E be the event that the first...

Suppose a fair coin is tossed 3 times. Let E be the event that the first toss is a head and let Fk be the event that there are exactly k heads (0 ≤ k ≤ 3). For what values of k are E and Fk independent?

Solutions

Expert Solution

ANSWER\

We first compute the probabilities for various events here as:
P(E) = P(first ross is a head) = 0.5 as first toss is equally likely to be a head or a tail.

P(F0) = P(no head in 3 tosses) = 0.53 = 0.125
P(F1) = P(1 head) = P(HTT, THT, TTH) = 3*0.125 = 0.375
P(F2 ) = P(2 heads) = P(HHT, HTH, THH) = 0.375,
P(F3) = P(3 heads) = P(HHH) = 0.125

Now, P(E and F0) = 0 as there is no outcome when both F0 and E happens. As this is not equal to P(E)P(F0) , therefore these 2 events are not independent.

P(E and F1) = P(HTT) = 0.125 but P(E)P(F1) = 0.375*0.5 = 0.1875 which is not equal to P(E and F1), therefore E and F1 are not independent here.

P(E and F2) = P(HHT, HTH) = 0.25 but P(E)P(F2) = 0.375*0.5 = 0.1875 which is not equal to P(E and F2), therefore E and F2 are not independent here.

P(E and F3) = P(HHH) = 0.125 but P(E)P(F3) = 0.125*0.5 = 0.0625 which is not equal to P(E and F3), therefore E and F3 are not independent here.

Therefore there is no value of k for which E and Fk would be independent.

If you have any doubts please comment and please don't dislike.

PLEASE GIVE ME A LIKE. ITS VERY IMPORTANT FOR ME.


Related Solutions

Suppose a fair coin is tossed 3 times. Let E be the event that the first...
Suppose a fair coin is tossed 3 times. Let E be the event that the first toss is a head and let Fk be the event that there are exactly k heads (0 ≤ k ≤ 3). For what values of k are E and Fk independent?
A fair coin is tossed, and a fair die is rolled. Let H be the event...
A fair coin is tossed, and a fair die is rolled. Let H be the event that the coin lands on heads, and let S be the event that the die lands on six. Find P(H or S).
A fair coin is tossed 3 times. Let X be equal to 0 or 1 accordingly...
A fair coin is tossed 3 times. Let X be equal to 0 or 1 accordingly as head or tail occurs on a first Toss. Let y equal to number of heads that occurs. Find A) the distribution of X and Y B) the joint distribution of X and Y C) whether they are independent or not D) the COV (XY)
A fair coin is tossed r times. Let Y be the number of heads in these...
A fair coin is tossed r times. Let Y be the number of heads in these r tosses. Assuming Y=y, we generate a Poisson random variable X with mean y. Find the variance of X. (Answer should be based on r).
A fair coin is tossed four times. Let X denote the number of heads occurring and...
A fair coin is tossed four times. Let X denote the number of heads occurring and let Y denote the longest string of heads occurring. (i) determine the joint distribution of X and Y (ii) Find Cov(X,Y) and ρ(X,Y).
A coin is tossed 400 times, landing heads up 219 times. Is the coin fair?
A coin is tossed 400 times, landing heads up 219 times. Is the coin fair?
Q7 A fair coin is tossed three times independently: let X denote the number of heads...
Q7 A fair coin is tossed three times independently: let X denote the number of heads on the first toss (i.e., X = 1 if the first toss is a head; otherwise X = 0) and Y denote the total number of heads. Hint: first figure out the possible values of X and Y , then complete the table cell by cell. Marginalize the joint probability mass function of X and Y in the previous qusetion to get marginal PMF’s.
Suppose I have tossed a fair coin 13 times and one of the following sequences was...
Suppose I have tossed a fair coin 13 times and one of the following sequences was my outcome: (1) HHHHHHHHHHHHH; (2) HHTTHTTHTHHTH; (3) TTTTTTTTTTTTT. Demonstrate that prior to actually tossing the coins, the three sequences are equally likely to occur (what is this probability?). What is the probability that the 13 coin tosses result in all heads or all tails? Now compute the probability that the 13 coin tosses result in a mix of heads and tails. Although the chances...
A fair coin is tossed two ​times, and the events A and B are defined as...
A fair coin is tossed two ​times, and the events A and B are defined as shown below. Complete parts a through d. ​A: {At most one tail is​ observed} ​B: {The number of tails observed is even​} a. Identify the sample points in the events​ A, B, Aunion​B, Upper A Superscript c​, and AintersectB. Identify the sample points in the event A. Choose the correct answer below. A. ​A:{TT comma HH​} B. ​A:{TT​} C. ​A:{HH comma HT comma TH​}...
A fair coin will be tossed three times. (a) Indicating a head by H and a...
A fair coin will be tossed three times. (a) Indicating a head by H and a tail by T write down the outcome space. (b) What is the probability that on the first toss the outcome with a tail? (c) What is the probability of obtaining exactly two heads from the three coin tosses? (d) What is the probability that the first toss gives a tail and exactly two heads are obtained from the three coin tosses? Are the outcomes...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT