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Category Archives: Normal/CLT approximation
Rock Paper Scissors
A game of rock paper scissors consists of several rounds (players continue to play rounds until one player wins). In one round of rock paper scissors, two players each choose one of three options (rock, paper, or scissors). If they choose two different options, the game ends (rock beats scissors, scissors beat paper, and paper beats rock). If both players choose the same option, the game continues for another round. Assume each player chooses rock, paper or scissors uniformly at random and independently.
- What is the distribution of number of rounds played in a single game?
- What is the expected number of rounds in a game? What is the standard deviation of number of rounds in a game?
- Let \(n\) be the number of games played. How big must \(n\) be to ensure at least 100 rounds are played with 90% probability? Use an appropriate approximation to estimate.
Ant crawls along a number line
An ant is crawling on a number line. The ant starts out at position \(0\). Every second the ant either
- Moves to the right 1 unit, with probability 1/2,
- Moves to the left 1 unit, with probability 1/4, or
- Stays at its current location, with probability 1/4
The ant’s movement during a particular second is independent of the ant’s previous movements. Let \(X_{160}\) be the ant’s location after 160 seconds.
- In 160 seconds, what is the probability that the ant moves to the right exactly 80 times, and to the left exactly 40 times?
- What is \(\mathbb{E}(X_{160})\)?
- What is \(Var(X_{160})\)?
- Let \(\mu=\mathbb{E}(X)\). Estimate \(\mathbb{P}(|X-\mu|\geq 15)\).
Working Batteries
A warehouse stores batteries. Most of the batteries work properly, but about 0.1%$ are faulty.
If a company orders 500 batteries, what is the probability that less than 3 will be faulty? Do this problem three ways:
- Find the probability exactly.
- Use a Poisson approximation to estimate.
- Use a normal approximation to estimate.
A company needs 10,000 working batteries. How many batteries should the company order from the warehouse in order to be 99.7% certain that they will receive at least 10,000 working batteries?
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 \(p\)?
August Birthdays
About 9% of birthdays (in the US) are in August. A researcher samples 10,000 people from the US and asks for their birthdays. Estimate the probability that between 850 and 950 of those people were born in August.
Approximating sums of uniform random variables
Suppose \(X_1,X_2,X_3,X_4\) are independent uniform \((0,1)\) and we set \(S_4=X_1+X_2+X_3+X_4\). Use the normal approximation to estimate \(\mathbf{P}( S_4 \geq 3) \).
Overloading an Elevator
A new elevator in a large hotel is designed to carry about 30 people, with a total weight of up to 5000 lbs. More that 5000 lbs. overloads the elevator. The average weight of guests at the hotel is 150 lbs., with a standard deviation of 55 lbs. Suppose 30 of the hotel’s guests get into the elevator . Assuming the weights of the guests are independent random variables, what is the chance of overloading the elevator ? Give your approximate answer as a decimal.
[Pitman p 204, # 19]
Random Digit
Let \(D_i\) be a random digit chosen uniformly from \(\{0,1,2,3,4,5,6,7,8,9\}\). Assume that each of the \(D_i\) are independent.
Let \(X_i\) be the last digit of \(D_i^2\). So if \(D_i=9\) then \(D_i^2=81\) and \(X_i=1\). Define \(\bar X_n\) by
\[\bar X_n = \frac{X_1 + \cdots+X_n}{n}\]
- Predict the value of \(\bar X_n \) when \(n\) is large.
- Find the number \(\epsilon\) such that for \(n=10,000\) the chance that you prediction is off by more than \(\epsilon\) is about 1/200.
- Find approximately the least value of \(n\) such that your prediction of \(\bar X_n\) is correct to within 0.01 with probability at least 0.99 .
- If you just had to predict the first digit of \(\bar X_{100}\), what digit should you choose to maximize your chance of being correct, and what is that chance ?
[Pitman p206, #30]
Basic Random Walk
Consider the following “game”: A marker is placed on the real line at the point zero. On each turn a coin is flip which a 1 printed on one side and a -1 printed on the other. If the 1 side lands face up, the marker is moved on unit in the positive direction while if the -1 lands heads up then the marker is moved one unit in the negative direction. If the coin has a probability of \(p\) of landing with the 1 side face up, answer the following questions:
- Let \(p=\frac12\). After 10000 turns if you had to pick one site to find the marker which would you choose ?
- Again let \(p=\frac12\). What is the approximate chance that the marker is further then 100 units from this most likely point after 10000 turns ? What is the approximate chance that the marker is further then 300 units from this most likely point after 10000 turns ?
- Repeat the above questions with \(p=\frac{9}{10}\).
