Question

Consider two coins, one fair and one unfair. The probability of getting heads on a given flip of the unfair coin is 0.10. You are given one of these coins and will gather information about your coin by flipping it. Based on your flip results, you will infer which of the coins you were given. At the end of the question, which coin you were given will be revealed.

When you flip your coin, your result is based on a simulation. In a simulation, random events are modeled in such a way that the simulated outcomes closely match real-world outcomes. In this simulation, each flip is simulated based on the probabilities of obtaining heads and tails for whichever coin you were given. Your results will be displayed in sequential order from left to right.

Here’s your coin! Flip it 10 times by clicking on the red FLIP icons:

Tails = 8

Heads = 2

What is the probability of obtaining exactly as many heads as you just obtained if your coin is the fair coin?

0.0013

0.1357

0.9453

0.0439

What is the probability of obtaining exactly as many heads as you just obtained if your coin is the unfair coin?

0.1937

0.1357

0.0013

0.9453

When you compare these probabilities, it appears more likely that your coin is the coin.

If you flip a fair coin 10 times, what is the probability of obtaining as many heads as you did or less?

0.0547

0.1357

0.0689

0.0013

The probability you just found is a measure of how unusual your results are if your coin is fair. A low probability (0.10 or less) indicates that your results are so unusual that it is unlikely that you have the fair coin; thus, you can infer that your coin is unfair.

On the basis of this probability, you infer that your coin is unfair.

Click here to find out whether you were flipping the fair coin or the unfair coin.

Answer #1

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Conducting a Simulation
For example, say we want to simulate the probability of getting
“heads” exactly 4 times in 10 flips of a fair coin.
One way to generate a flip of the coin is to create a vector in
R with all of the possible outcomes and then randomly select one of
those outcomes. The sample function takes a vector of elements (in
this case heads or tails) and chooses a random sample of size
elements.
coin <- c("heads","tails")...

A selection of coin is known to be either fair (with a
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An unfair coin is such that on any given toss, the probability
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0.4. The coin is tossed 8 times. Let the random variable X be the
number of times heads is tossed.
1. Find P(X=5).
2. Find P(X≥3).
3. What is the expected value for this random variable?
E(X) =
4. What is the standard deviation for this random variable? (Give
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SD(X)...

Flip 2 fair coins. Then the sample space is {HH, HT, TH, TT}
where T = tails and H = heads. The outcomes HT & TH are
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A black bag contains two coins: one fair, and the other biased
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Consider the experiment of flipping 3 coins. Determine the
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A) what is the probability that the second coin is
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B) what is the probability that the second coin is heads given
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Part 1. Two fair coins are tossed and we are told that one
turned up “Heads”. What is the probability that the other turned up
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Part 2. Two fair coins are tossed, and we get to see only one,
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*Remark This problem is really different from the previous
one!

You flip a coin until getting heads. Let X be the number of coin
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