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Category Archives: Coin Flips
Consecutive Tails
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 \(\mathbf{P}(A_{21})\)
Flipping Coins and Independence
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?
Biased coin
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, \(p=\mathbb{P}(Heads).\)
- Suppose \(p=0.8\). What is the probability of observing between 76 and 84 heads out of 100 flips of the coin.
- Suppose you flip the coin 100 times and observe 80 heads. What is the 95% confidence interval for \(p\)?
Coin flipping game
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.
- Compute \(\mathbb{P}(A)\).
- Are \(A\) and \(B\) independent?
- Are \(A\) and \(C\) independent?
The chance a coin is fair
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.
- What is the probability that I have thrown the fair coin ?
- If I throw the same coin again, and heads comes up again, what is the probability that I have thrown the fair coin ?
- If instead of throwing the same coin again, I reach into my pocket and throw the second coin. If it comes up heads, what is the chance the first coin is the fair coin ?
[ Modified version of Meester, ex 1.7.35]
Making a biased coin fair
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 ?
Coin tosses: independence and sums
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.
- Make a table showing the joint distribution of \(X\) and \(Y\).
- Are \(X\) and \(Y\) independent ?
- Find the distribution of \(X+Y\) ?
Picking a box then a ball
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.
- What is the probability that a 3 is chosen ?
- What is the probability a number less than or equal to 2 is chosen ?
- The above procedure produces a distribution on \(\{1,2,3,4,5\}\) how does it compare to picking a number uniformly (with equal probability) ?
[Pitman p 37, example 5]
Betting with Coin Flips
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.
- What is the probability that Bob and Alice have exactly the same amount of money after \(2n\) flips ?
- What is the chance that Alice has more money after \(2n+1\) flips ?
Introduction to Geometric random variables
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]
Duels
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) ?
Coin Flips: typical behavior
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.
- Without calculations, say which of these outcomes is more likely. Why ?
- Confirm your answer to the previous question by a calculation.
[Pitman, p. 108 #3]
