```M203E Practice Quiz 4 Solutions

[1] Say you roll a fair die twice.
(a) What's the chance that the first time the die comes up 2 or more?
There are 36 outcomes, 6 possibilities for the first roll, and for
each of those 6, there are 6 for the second roll. Five of the 6
outcomes of the first roll consist of the die coming up 2 or more,
so there are 5x6 outcomes for two rolls where the first roll is a 2
or more. Thus the chance of the first roll being a 2 or more is
5x6/(6x6) = 5/6.

(b) What's the chance that the first time the die comes up 2 and the
second time the die comes up 5?

The chance is 1/36.

(c) What's the chance that either the first time the die comes up 2
or the second time the die comes up 5?

Here are the 6 outcomes where you get a 2 on the first roll:
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6).
Here are the 6 outcomes where you get a 5 on the second roll:
(1,5), (2,5), (3,5), (4,5), (5,5), (6,5)
Thus there are 11 outcomes where you get either a 2 on the first roll
or a 5 on the second roll (not 12, since you can't count (2,5) twice).
Thus the chance is 11/36.

[2] A bad of marbles has 3 red marbles and 5 blue marbles.
(a) If you reach into the bag and pick a marble at random,
what's the chance it is red?

The chance is 3/8.

(b) If you reach into the bag twice and pick a marble each time
(without replacing the marbles you pick), what's the chance
the marbles have different colors?

There are 8x7 ways to pick two marbles. Of these, there are 3x5 ways
for the first marble to be red and the second marble to be blue,
and there are 5x3 ways for the first marble to be blue and the
second marble to be red. Thus the chance is (3x5 + 5x3)/(8x7) = 30/56.

(c) If you reach into the bag and pick a marble, record the color,
and replace it back in the bag, and then reach into the bag and pick a
marble again, what's the chance now that the marbles have different colors?

There are now 8x8 ways to pick two marbles, because now you could pick
the same marble twice, but there are still just 30 ways, as before, to pick
marbles of different colors. So the chance is 30/64.

[3] A fisheries official tags and releases 100 fish into a pond.
After a short period of time (to allow the tagged fish to mix in
with the whole fish population in the pond), she catches a sample
of 20 fish, of which 4 are tagged. What does she estimate the
pond's fish population to be?

Experimentally the chance of a random fish having a tag is 4/20.
But if N is the total fish population, the mathematical chance
of a random fish having a tag is 100/N. Thus 4/20 = 100/N,
so N = 100x20/4 = 500.

[4] In February 2006 a group of eight people from Lincoln won a jackpot valued at
177 million dollars. In an Omaha World Herald column
(Feb 26, 2006), Harold Andersen (whom the journalism building here at
UNL is named after) wrote: ``The Lottery Commission's intent seems pretty obvious:
Encourage more gambling by Nebraskans, in spite of the fact that the law of
probabilities makes it less likely that a record-breaking Powerball payoff
will strike in Nebraska again.'' Is there such a law
of probabilities? If you flip a fair coin three times and it comes up
heads each time, is it more likely to come up heads or tails the
next time you flip it? Justify your answer to the second question
and then indicate whether you agree or disagree with Andersen's
assertion that a second payoff is less likely as a result of there
having been a payoff recently.

The probability of a fair coin coming up heads doesn't change, even if
it's come up heads several times recently. It's true that in the long run
we expect the average rate of heads to be about 50% (this is the law
of probabilities that Andersen is referring to, sometimes called the
Law of Large Numbers), but this is what naturally happens if heads
and tails are equally likely. There's nothing that forces some tails
to happen to make up for a few heads. For example, if half the population
has brown hair, and if one day you see several brown haired people
on the street, that doesn't mean that all of a sudden it becomes more
likely to see a person with blonde hair. Regarding the lottery payoff,
the chance of a win in Lincoln does not change just because Lincoln
has had a winner recently. In fact, after a big payoff in Lincoln,
the chance of another win in Lincoln is probably more than before,
because more people in Lincoln may buy tickets from the store that
sold the winner, on the false premise of that store being "lucky".
The more Lincolnites that buy tickets, the more likely it will be
for someone from Lincoln to win!
```