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Assignment3.pdf-Stat 241/251: Assignment 3 Due Monday
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Assignment3.pdf-Stat 241/251: Assig...
Assignment3.pdf-Stat 241/251: Assignment 3 Due Monday
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Stat 241/251: Assignment 3
Due Monday March 21
st
by 8:50am
1. Suppose you work as a waiter/waitress at Mahoney & Sons Pub on UBC campus.
From your time working there you know that the average tip (per table) you receive is
\$7.11 with a standard deviation of \$5.90. The distribution for the amount of tip you
receive is highly skewed to the right.
(a) What is the probability that a table tips you more than \$13.50? If this question
cannot be answered with the information given, state why.
(b) What is the probability that the average tip you receive from 5 randomly selected
tables you wait on is more than \$13.50? If this question cannot be answered with
the information given, state why.
(c) On a typical weekend, you usually wait on 30 tables over the entire weekend.
What is the probability that on a given weekend (assuming you wait on exactly
30 tables) that you make more than \$250 in tips?
If this question cannot be
answered with the information given, state why.
2. When using the Google search engine, one can see the amount of time that was taken
for the search engine to provide you with a list of pages.
Suppose that the amount
of time until a list of pages is provided is normally distributed with a mean of 0.33
seconds and a standard deviation of 0.08 seconds.
(a) What is the probability that a randomly selected search takes more than 0.4
seconds?
(b) What is the 95
th
percentile for search times?
(c) Now, consider a set of 70 searches using the Google search engine. What is the
probability that at most, 10 of the 70 searches take longer than 0.4 seconds? You
may use an approximate method.
(d) Again, consider a set of 70 searches.
What is the probability that the mean
amount of time for the 70 searches is more than 0.34 seconds?
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3. In this question we will examine the casino game Roulette.
Essentially, the game
involves randomly selecting a number, from 38 possible numbers.
There are 18 red
numbers, 18 black numbers and 2 green numbers. All numbers are equally likely to
show up on a given spin, and spins of the wheel are independent. Consider making
bets on the colour of the number coming up in Roulette (red or black). This bet pays
1:1, meaning if you bet \$1, winning means you gain \$1 and losing means you lose \$1.
(a) If you make 50 bets, each of \$1, what is the probability that you leave the casino
having made money? That is, you leave the casino with more money than you
started with. Use an approximate method.
(b) If you make 500 bets, each of \$1, what is the probability you leave the casino
having made money? Use an approximate method.
(c) If you make 5000 bets, each of \$1, what is the probability you leave the casino
having made money? Use an approximate method.
(d) Sketch out a plot representing the probability of making money (y) as a function
of the number of bets made (x). You don’t have to calculate any more than you
already have; you may approximate the shape of this curve.
Brieﬂy comment
on what happens to the probability of making money as the number of bets (n)
(e) Consider again making 5000 bets, each of \$1, on the colour of the number in
Roulette. Create an interval for the amount won (or lost) that you will fall into
95% of the time. Center this interval around the expected loss. Comment on this
interval.
4. Suppose that small airplanes arrive at a small airplane airport at a rate of 4 per hour,
and are assumed to follow a poisson process.
(a) What is the probability that at most 3 planes land in a 75 minute period?
(b) What is the probability that upon the airport opening for the day, it takes more
than 30 minutes for the ﬁrst plane to arrive?
(c) Consider an entire day. We will assume that a day consists of 16 hours of operation
for the airport.
What is the probability that at least 50 planes land over the
one day (16 hour) period? You may use an approximate method to answer this
question.
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