Let \(A_n\) be the event that in \(n\) flips of a fair coin, there are never 2 consecutive tails. Suppose we know the following probabilities.
- \(\mathbf{P}(A_{19})\approx 0.021\)
- \(\mathbf{P}(A_{20})\approx 0.017\)
Evaluate \(\mathbf{P}(A_{21})\)
Let \(A_n\) be the event that in \(n\) flips of a fair coin, there are never 2 consecutive tails. Suppose we know the following probabilities.
Evaluate \(\mathbf{P}(A_{21})\)
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Posted in Coin Flips, Conditioning
An experimenter has two fair coins and one biased coin. The biased coin lands on heads with probability 3/4.
The experimenter randomly selects one of the three coins and flips it until they get heads.
Let \(A\) be the event that the experimenter flipped the biased coin.
Let \(B\) be the event that it took the experimenter an even number of flips to get heads.
Are events \(A\) and \(B\) independent?
Comments Off on Flipping Coins and Independence
Posted in Coin Flips, Geometric Distribution, Independence, Series
You have a biased coin, but you don’t know what the bias is. Let \(p\) be the actual probability of getting heads on a single coin flip, \(p=\mathbb{P}(Heads).\)
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Posted in Binomial, Coin Flips, Confidence Interval, Normal/CLT approximation
Your friend challenges you to a game in which you flip a fair coin until you get heads. If you flip an even number of times, you win. Let \(A\) be the event that you win. Let \(B\) be the event that you flip the coin 3 or more times. Let \(C\) be the event that you flip the coin 4 or more times.
Comments Off on Coin flipping game
Posted in Coin Flips, Geometric Distribution, Independence, Series
Suppose that I have two coins in my pocket. One ordinary, fair coin and one coin which has heads on both sides. I pick a random coin out of my pocket, throw it, and it comes up heads.
[ Modified version of Meester, ex 1.7.35]
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Posted in Bayes Theorem, Coin Flips, Conditioning
Jack and Jill want to use a coin to decide who gets the remaining piece of cake. However, since the coin is Jack’s, Jill is suspicious that the coin is a trick coin which produced head with a probability \(p\) which is not \(\frac12\). Can you devise a way to use this coin to come to a fair decision as to who gets the cake ?
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Posted in Coin Flips
A fair coin is tossed three times. Let \(X\) be the number of heads on the first two tosses, \(Y\) the number of heads on the last two tosses.
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Posted in Coin Flips, Independence, Sequence of independent trials
Tagged JCM_math230_HW4_S13, JCM_math230_HW5_S15, JCM_math340_HW4_F13
Suppose that there are two boxes, labeled odd and even. The odd box contains three balls numbered 1,3,5 and the even box contains two balls labeled 2,4. One of the boxes is picked randomly by tossing a fair coin.
[Pitman p 37, example 5]
Posted in Basic probability, Coin Flips, Conditioning, Drawing Balls
Alice and Bob flip a coin repeatedly. Each time there is a head bob gets a dollar and each time there is a tail Alice gets a dollar.
Posted in Coin Flips, Describing State spaces
Consider flipping a coin that is either heads (H) or tails (T), each with probability 1/2. The coin is flipped over and over (independently) until a head comes up. The outcome space is
\[ \Omega = \{H,TH,TTH,TTTH,\ldots\}. \]
(a) What is \( \mathbf{P}(TTH)\)?
(b) What is the chance that the coin is flipped exactly \(i\) times?
(c) What is the chance that the coin is flipped more than twice?
(d) Repeat the previous three questions for a unfair coin which has probability \(p\) of getting Tails.
[Author Mark Huber. Licensed under Creative Commons]
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Posted in Coin Flips, Geometric Distribution
Tagged JCM_math340_HW4_F13
Mathematicians and politicians throughout history have dueled.
Alexander Hamilton and Aaron Burr dueled.
The French mathematician Evariste Galois died in a duel.
Consider two individuals (H) and (B) for example dueling.
In each round they simultaneously shoot the other and the probability
of a fatal shot is \(0 < p < 1\).
1) What is the probability they are fatally injured in the same round ?
2) What is the probability that (B) will be fatally injured before (H) ?
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Posted in Coin Flips, Conditioning, Sequence of independent trials
A fair coin is tossed repeatedly. Considering the following two possible outcomes:
55 or more heads in the first 100 tosses.
220 or more heads in the first 400 tosses.
[Pitman, p. 108 #3]
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Posted in Coin Flips, Typical vs Atypical
Tagged JCM_math230_HW3_F22, JCM_math230_HW3_S13, JCM_math230_HW3_S15, JCM_math340_HW2_F13