Peter McFetridge, 25480138
Jan 22, 2016
Math 302
1. Section 1.2 Exercise 6
For any there is a distribution function where the probability of side facing up, is proportional to the number of dots on side .
or
Let be the event that an even number is thrown. So,
2. Section 1.2 Exercise 7
Let A and B be events such that , , and . What is ?
$$P(A) = 1 - P(\tilde{A})$ then $
And
Gives us
3. Section 1.2 Exercise 10
In order for a bill to get to the president it must pass the House of Representatives and the Senate.
Let a bill passing the Senate
Let a bill passing the House of Representatives
The probability that a bill is passed throught the Senate is and the probability that a bill is passed through the house is and the probability that a bill is passed through at least one of the two is
Since
Then
So,
4. Section 1.2 Exercise 11
What odds should a person give in favor of the following events?
(a) A card chosen at random from a 52 card deck is an ace.
= ace is drawn There are 4 aces in a 52 card deck
(b) Two heads will turn up when a coin is tossed twice.
= coin toss comes up heads We want the probability that we will receive 2 heads or
Probability that two heads will turn up =
This can also so seen by looking at the total number of outcomes. In this case there are 4 total outcome.
Only 1 of the 4 outcomes satisfies what we are looking for and therefore
(c) Two sixes will turn up when two dice are rolled.
Omega is the sample space of the results when a die is roll with an even distribution function. Event = 6 facing up when die is rolled. We are looking for the probability that both die are 6
5. Section 1.2 Exercise 18
(a) For events , prove that
Let are events in the set
Since is the event that at least one event in occurs and is bounded by by Axiom 2. Regardless if the events are distinct or not distinct.
By Axiom 1 then it is easy to see that unless event .
(b) For events A and B, prove that
Let and be events in
We can rearrange the problem Which is equivalent to Since we have Which is true by Axiom 2
6. Section 1.2 Exercise 18
If are any three events, show that
We can expand
7. Section 1.2 Exercise 20
Explain why it is not possible to define a uniform distribution function (see Definition 1.3) on a countably infinite sample space. Hint: Assume m(ω) = a for all ω, where 0 ≤ a ≤ 1. Does m(ω) have all the properties of a distribution function?
For
We can show that
Which is true for the geometric sequence, or
for
When
Which shows that and
And that contradicts the properties of a uniform distribution.
8. Section 1.2 Exercise 26
Two cards are drawn from a deck of 52 cards. What is the probability that the second card is higher in rank than the first card.