Question

In: Statistics and Probability

Given that in 4 flips of a fair coin there are at least two "heads", what...

Given that in 4 flips of a fair coin there are at least two "heads", what is the probability that there are two "tails"? There are ten equally likely outcomes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. You randomly select one value, call it the initial value. Then, you continue to randomly select values, call them follow-up selections, until you come up with the initial value. What is the fewest number of follow-up selections that insures that the probability is better than one-half that you will observe your initial value? (Note: this problem assumes values are selected "with replacement," which simply means that after each selection, there are still the same ten equally likely outcomes.)

Expert Solution

The the number of tails/heads out of 4 flips of a fair coin has Binomial distribution with
The PMF of is .

The probability of at least two "heads" is

The probability that there are two "tails"

The number of follow-up selections needed to observe the initial value has geometric distribution.

The PMF is

The probability that you will observe your initial value in selections is

We need,

$1-0.9^n>0.5\\ 0.9^n<0.5$

Taking logarithms on both sides (Here logarithms results in negative value. So the inequality sign reverses after division by negative number)

The minimum number of follow-up selections is .

orchestra answered 2 years ago

There is a fair coin and a biased coin that flips heads with probability 1/4.You randomly...

There is a fair coin and a biased coin that flips heads with probability 1/4.You randomly pick one of the coins and flip it until you get a heads. Let X be the number of flips you need. Compute E[X] and Var[X]

In a sequence of independent flips of a fair coin thwr comes up heads with probability...

In a sequence of independent flips of a fair coin thwr comes up heads with probability 0.6, what is the probability that there is a run of three consecutive heads within the first 10 flips? Show a Markov chain that counts the number of consecutive heads attained.

What’s the probability that a fair coin results in 12 or fewer heads from 40 flips?...

What’s the probability that a fair coin results in 12 or fewer heads from 40 flips? And What does that probability say about the fairness of the coin?

Bob flips a coin 3 times. Let A be the event that Bob flips Heads on...

Bob flips a coin 3 times. Let A be the event that Bob flips Heads on the first two flips. Let B be the event that Bob flips Tails on the third flip. Let C be the event that Bob flips Heads an odd number of times. Let D be the event that Bob flips Heads at least one time. Let E be the event that Bob flips Tails at least one time. (a) Determine whether A and B are...

2. Bob flips a coin 3 times. Let A be the event that Bob flips Heads...

2. Bob flips a coin 3 times. Let A be the event that Bob flips Heads on the first two flips. Let B be the event that Bob flips Tails on the third flip. Let C be the event that Bob flips Heads an odd number of times. Let D be the event that Bob flips Heads at least one time. Let E be the event that Bob flips Tails at least one time. (a) Determine whether A and B...

You roll two fair four-sided dies and then flip a fair coin. The number of flips...

You roll two fair four-sided dies and then flip a fair coin. The number of flips is the total of the roll. a. Find the expected value of the number of heads observed. b. Find the variance of the number of heads observed.

Two successive flips of a fair coin is called a trial. 100 trials are run with...

Two successive flips of a fair coin is called a trial. 100 trials are run with a particular coin; on 22 of the trials, the coin comes up “heads” both times; on 60 of the trials the coin comes up once a “head” and once a “tail”; and on the remaining trials, it comes up “tails” for both flips. Is this sufficient evidence ( = : 05) to reject the notion that the coin (and the flipping process) is a...

Flip a fair coin 4 times. Let ? and ? denote the number of heads and...

Flip a fair coin 4 times. Let ? and ? denote the number of heads and tails correspondingly. (a) What is the distribution of ?? What is the distribution of ? ? (b) Find the joint PMF. Are ? and ? independent? (c) Calculate ?(? ?) and ?(X≠?)(d) Calculate C??(?, ? ) and C???(?, ? )

Fair Coin? A coin is called fair if it lands on heads 50% of all possible...

Fair Coin? A coin is called fair if it lands on heads 50% of all possible tosses. You flip a game token 100 times and it comes up heads 41 times. You suspect this token may not be fair. (a) What is the point estimate for the proportion of heads in all flips of this token? Round your answer to 2 decimal places. (b) What is the critical value of z (denoted zα/2) for a 99% confidence interval? Use the...

You try to find out whether a coin is fair or has two heads. The coin...

You try to find out whether a coin is fair or has two heads. The coin is tossed three times, after which you will be shown the outcome of all three coin tosses. If you decide to conclude that the coin is fair if it shows heads and tails at least once, and unfair if all three tosses are heads, what is the probability of committing a type I error and a type II error?