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

1. & 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 .
2. !! Division into Cases.
   • General formula:
   • We seek , use
   • Use and therefore
   • So we use:
3. && Plug in data and compute the answer.
   • Know
   • Know
   • Know
   • Know
   • Therefore:

11 - Inferring bin from marble

1. & 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
2. && Identify knowns.
   • Know
   • Know
   • Know
   • Know
   • Know
3. !! Translate Bayes’ Theorem
   • Bayes’ Theorem for :
   • Division into Cases for the denominator:
4. && Plug in data and compute the answer.
   • Denominator:
   • Desired event:

12 - Independence and complements

(1) We show that

1. &&& Assume and show .
   • Divide into the cases:
   • Apply the assumption:
   • Algebra:
   • Negation rule:
2. & Assume and show
   • !! In the above sequence, apply this assumption to break up the second term instead.

(2)

1. & 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:

17	4	4	4	5
So we have:

Expectation and variance

27 - Gambling game - tokens in bins

1. & Setup.
   • Let be a random variable measuring your winnings in the game.
   • The possible values of are 1, 50, and 1000.
2. && Find PDF .
   • For have
   • For have
   • For have
   • These add to 1, and for all other .
3. && Find .
   • Using the discrete formula:
4. & 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)