Probability
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Exercise solutions - Unit 01

Events and outcomes

02 - Coin flipping: counting subsets

There are possible sequences, so .

To count the number of possible subsets, consider that we have 32 distinct items, and a subset is uniquely determined by the binary information – for each item – of whether it is in or out. Thus there are possibilities. So .

Conditional probability

07 - Simplifying conditionals

    • Definition of ‘conditional’:
    • The problem assumes that . Therefore .
    • Therefore, answer: .
    • Definition of ‘conditional’:
    • Since , we know .
    • Therefore and answer .
    • Definition of ‘conditional’:
    • Since , we have .
    • Therefore, answer .
    • Definition of ‘conditional’:
    • There is no way to simplify further.
    • We could write if desired.

09 - Division into Cases

Label events.

Event : a red marble is transferred

Event : a green marble is transferred

Event : a red marble is drawn from Bin 2

Event : a green marble is drawn from Bin 2

Answer will be .

Division into Cases.

General formula:

We seek , use

Use and therefore

So we use:

Plug in data and compute the answer.

Know

Know

Know

Know

Therefore:

11 - Inferring bin from marble

Label events.

Event : friend chooses Bin 1

Event : friend chooses Bin 2

Event : friend draws a red marble

Event : friend draws a green marble

Answer will be

Identify knowns.

Know

Know

Know

Know

Know

Translate Bayes’ Theorem

Bayes’ Theorem for :

Division into Cases for the denominator:

Plug in data and compute the answer.

Denominator:

Desired event:

12 - Independence and complements

(1) We show that

Assume and show .

Divide into the cases:

Apply the assumption:

Algebra:

Negation rule:

Assume and show

In the above sequence, apply this assumption to break up the second term instead.

(2)

Show that and are equivalent.

In the first equivalence, replace with and with . Use too.

Counting

15 - Counting teams with Cooper

There are teams that include Cooper, and teams in total. So we have:

18 - Counting out 3 teams

This is just the multinomial coefficient with this data:

174445

So we have:

Expectation and variance

27 - Gambling game - tokens in bins

Setup.

Let be a random variable measuring your winnings in the game.

The possible values of are 1, 50, and 1000.

Find PDF .

For have

For have

For have

These add to 1, and for all other .

Find .

Using the discrete formula:

Conclusion

Since , if you play it a lot at $50 you will generally make money.

Challenge Q:

If you start with $200 and keep playing to infinity, how likely is it that you go broke?

Function on a random variable

36 - Probabilities via CDF

(a)

(b) Same as (a) because (single point in a continuous distribution).

(c)

(d)