I have added to the list of Problem Solving Strategies a note about the inclusion-exclusion formula.
I showed you in Example 4.5 a use of the probabilistic method in combinatorics. If you liked that, you might like to read about other examples, for instance in this link.
In Example 4.4 we considered an infinite sequence of tosses of biased coins and saw that if ∑∞i=1pi<∞ then P(infinite number of heads)=0.
Now let's prove that ∑∞i=1pi=∞⟹P(finite number of heads)=0. This is trickier.
We are assuming the coin tosses are independent (which is terminology from Lecture 5, but you can guess intuitively what this means). Let Ak be the event that the kth toss is a head. Let En=⋂∞k=nAck=[no heads after the (n−1)th toss]. Then for m>n
P(En)≤P(⋂mk=nAck)=∏mk=nP(Ack)=∏mk=n(1−pk)≤∏mk=ne−pk=e−∑mk=npk.
The right hand side →0 as m→∞.
Hence P(En)=0, and then using Property (v) of P (continuity), we have that
P(finite number of heads)=P(⋃∞n=1⋂∞k=nAck)=limn→∞P(En)=0.
These two results are examples of applications of the Borel-Cantelli lemmas, which appear in the Part II course Probability and Measure.
I did not have time to talk through Example 4.8, but it is another nice use of the inclusion-exclusion formula.
I showed you in Example 4.5 a use of the probabilistic method in combinatorics. If you liked that, you might like to read about other examples, for instance in this link.
In Example 4.4 we considered an infinite sequence of tosses of biased coins and saw that if ∑∞i=1pi<∞ then P(infinite number of heads)=0.
Now let's prove that ∑∞i=1pi=∞⟹P(finite number of heads)=0. This is trickier.
We are assuming the coin tosses are independent (which is terminology from Lecture 5, but you can guess intuitively what this means). Let Ak be the event that the kth toss is a head. Let En=⋂∞k=nAck=[no heads after the (n−1)th toss]. Then for m>n
P(En)≤P(⋂mk=nAck)=∏mk=nP(Ack)=∏mk=n(1−pk)≤∏mk=ne−pk=e−∑mk=npk.
The right hand side →0 as m→∞.
Hence P(En)=0, and then using Property (v) of P (continuity), we have that
P(finite number of heads)=P(⋃∞n=1⋂∞k=nAck)=limn→∞P(En)=0.
These two results are examples of applications of the Borel-Cantelli lemmas, which appear in the Part II course Probability and Measure.
I did not have time to talk through Example 4.8, but it is another nice use of the inclusion-exclusion formula.
Confused · 584 weeks ago
$$mathcal{P}left(bigcap_{i=1}^infty bigcup_{k=i}^infty A_k right)$$, I don't remember your reasoning in lectures? Is it an infinite sequence of heads, or an infinite sequence of heads possibly preceded by some finite collection of heads and tails?
Richard Weber 35p · 584 weeks ago
I think it is worth writing a post about this.
So see http://weberprobability.blogspot.co.uk/2014/01/li...
Charlie · 583 weeks ago
Richard Weber 35p · 583 weeks ago