1. (a) If 48% of all teenagers own a skateboard and 39% of all teenagers own a
skateboard and roller blades. What is the probability that a teenager owns
roller blades given that the teenager owns a skateboard?
Hint: The conditional probability is given by
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P (A|B) = P (A, B)
(b) If 88% of all households have a television and 51% of all households have a
television and a DVD. What is the probability that a household has a DVD
given that it has a television?
(c) 84% of the houses have a garage and 65% of the houses have a garage and
garden. What is the probability that a house has garden given that it has a
2. The survival probabilities for men are as follows:
i) probability that a man lives at least 70 years: 80%
ii) probability that a man lives at least 80 years: 50%.
What is the probability that a man lives at least 80 years given that he has just
turned 70?
3. Which distribution of the ones discussed in lectures would you use to model each
of the following random variables?
(a) A woman is trying to hit the 20 on a dart board. Let X1 = the number of
throws before she hits the 20.
(b) X2 = the time between eruptions of a volcano.
(c) X3 = the number of eruptions of a volcano in 1000 years.
(d) X4 = the time between 3 eruptions of a volcano.
(e) A shop is open between 11 and 3 and customers arrive at any time at an
apparently constant rate. X5 = the arrival time of any particular customer.
(f) A woman is trying to measure the length of a deck with a tape measure. The
deck is 20m long. X6 = the measurements obtained by the woman.
(g) A woman likes only some of the songs in her Spotify that she is listening to
on shuffle. X7 = whether or not she likes the next song.
(h) A student sits 25 tests in a year and needs to get beyond a certain threshold
in each test to pass. X8 = Number of tests in which the threshold is achieved.
4. Memorylessness
Think of the number of tries it would take to crack a safe with 500 combinations. What does that distribution look like? How many does it take on average, and how many tries is the maximum?
Then think of the number of tries if you have an infinite line of safes, all with possibly different combinations, and you only get to try each one once.
Which of these processes is memoryless? The distribution it makes has the memoryless property. If you can rule out combinations, it matters how many combinations you’ve already tried, but with a new safe each time it doesn’t matter.
More formally, if a distribution is memoryless, then the probability of that random variable being greater than X+Y given we know it’s at least X, is the same as the probability of it being greater than Y given that its at least 0.
Draw a geometric distribution for how many coins flips it takes to flip heads.
Now draw one for how many flips it takes if you already flipped 10 tails.
Is the chance of flipping at least 1 more tails before a head different?
Draw a uniform distribution for the chance that you arrive at a shop during each hour, assuming you can arrive between 1 and 10pm and any time is equally likely. What is the probability that you arrive in the first hour after 1pm?
Now draw a uniform distribution assuming you haven’t arrived before 5pm. Given that, what is the probability you arrive in the first hour after 5pm?
Is the probability of going to the shop in the next hour different?
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