We model the experiment where we toss a fair coin 4 times as follows. The sample...

We model the experiment where we toss a fair coin 4 times as follows. The sample space Ω is the space of (ordered) binary vectors of length 4. A 1 in the i-th position of such a binary vector indicates Heads at the i-th toss. The probability measure (P) on Ω is defined as P(A)=|A| / |Ω| for all A⊂Ω. Let Ai be the event that the ith toss is Heads, i = 1,...,4. (a) Give all outcomes in A3. (b) Show that, with the above probability measure P, the events A1, . . . , A4 are mutually independent. (c) Give P(A1 ∩ A2 | A1) and P(A1 ∪ A3 | A2)

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orchestra answered 2 years ago

Suppose you toss a fair coin 4 times. Denote the outcome to be 1 if you...

Suppose you toss a fair coin 4 times. Denote the outcome to be 1 if you get a head and 0 if a tail. a) Write down the sample space Ω b) What is the probability of the event that you get head at least once? c) If you get four same toss you will get 10 dollars, otherwise you will lose 2 dollars. On average, will you win or lose?

Using R-studio 2. Consider an experiment where we flip a fair coin six times in a...

Using R-studio 2. Consider an experiment where we flip a fair coin six times in a row, and i is the number of heads tossed: a. Calculate the probability mass function for i = 0. . . 6 using the equation from Ross section 2.8 for Binomial Random Variables b. Conduct a simulation of this experiment in R, with T trials of the experiment – pick several values of T from 10 to 10,000. c. Create a...

we toss a fair coin 100 times. What is the probability of getting more than 30...

we toss a fair coin 100 times. What is the probability of getting more than 30 heads?

If I toss a fair coin 50,000 times which of the following is true? a) the...

If I toss a fair coin 50,000 times which of the following is true? a) the number of heads should be between 15,000 and 25,000. b) the proportion of heads should be close to 50%. c) the proportion of heads in these tosses is a parameter. d) the number of heads should be exactly 25,000. e) the proportion of heads will be close to 1.

Consider the following experiment: Simultaneously toss a fair coin and, independently, roll a fair die. Write...

Consider the following experiment: Simultaneously toss a fair coin and, independently, roll a fair die. Write out the 12 outcomes that comprise the sample space for this experiment. Let X be a random variable that takes the value of 1 if the coin shows “heads” and the value of 0 if the coin shows “tails”. Let Y be a random variable that takes the value of the “up” face of the tossed die. And let Z = X + Y....

(1) For the random experiment “toss a fair coin twice and then roll a balanced die...

(1) For the random experiment “toss a fair coin twice and then roll a balanced die twice”, what’s the total count of the sample space? Let A be the event {both tosses show HEADs and both rolls show sixes}, B = {both tosses show the same side of the coin and both rolls show the same number}, C = {both tosses show the same side OR both rolls show the same number} and D = {the 1st roll of the...

A fair coin is tossed 4 times. a. Write the outcomes of the sample space b....

A fair coin is tossed 4 times. a. Write the outcomes of the sample space b. Let A be the event of obtaining at least one head, find P(A) c. Let B be the event of obtaining at least one tail, find P(B) d. Let C be the event of obtaining two tails, find P(C) e. Find P(A ∪ B) f. Find P(A ∩ B) g. Find P(A ∪ B ∪ C)

I toss a fair coin 20 times. (a) Calculate the probability of tossing 18 or more...

I toss a fair coin 20 times. (a) Calculate the probability of tossing 18 or more heads exactly. (b) Now perform the same calculation, approximating the actual binomial distribution with a normal distribution, picking a proper random variable, and using the correct mean and variance. (c) Do the results reasonably agree?

Suppose you toss a fair coin 10 times resulting in a sequence of heads (H) and...

Suppose you toss a fair coin 10 times resulting in a sequence of heads (H) and tails (T). Let X be the number of times that the sequence HT appears. For example, HT appears thrice in THTHHTHHHT Find E(X). Use Indicator random variables.

What are the probabilities that... 1) You toss a fair coin 6 times and you get...

What are the probabilities that... 1) You toss a fair coin 6 times and you get exactly 4 heads? 2) You toss a fair coin 6 times and you get at least two heads?

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