| Used to find the probability of the first successful outcome from an infinite number of trials
characteristics
Infinite number of trials
Constant Probability (p)
only two possible outcomes (yes/no, true/false etc)
Independent trials
| This is an interesting one. It is really very simple but it is important to learn the method .
Say for example three boys toss a coin taking it is turns until one of them gets a "head".
What is the probabilty that the third boy to toss the coin will get the first head.
Looking at it logically -
the first boy will have a 50%, or 0.5 or half chance of getting a head and winning.
The second boy will also have a 0.5 chance of getting a head
BUT will obviously only get a go if the first boy failed.
So his chance of having a go is only 0.5 and then he has a 0.5 chance of getting a head,
so he has a 0.5 * 0.5 = 0.25 chance of winning.
The third boy will only get a go if the first two have failed
so his chance of getting a go is 1 - 0.5 - 0.25 = 0.25
His chance of getting a head is the same as the others, 0.5
So his chance of winning is 0.25 * 0.5 = 0.125
All well and good BUT what if they all fail on their first attempt?
We have to start again with the first boy and so on.
What initially looked like an easy problem to solve suddenly got very tedious.
Luckily we have a method to help a bit.
P(X) = (1-p)x-1 * p
where x is the sequence number of the trial (1=1st attempt, 2 = 2nd attempt etc)
and p is the probability of success
So in the above example - the probability of the third boy getting a head on his first toss is :
P(3) = (1 - 0.5)3-1 * 0.5
= (0.5)2 * 0.5
= (0.25) * 0.5 = 0.125
Phew - same answer as the long way!!
Of course if we want to know the probability of the 3rd boy winning then we must do:
P(3,6,9,12 ....) in order to cover all his chances - to infinity and beyond
(I made up the last bit there - just to infinity will do)
In fact it is usually enough just to do Three terms - but the examiner will tell you what to do
(pretty hard to go to infinity in the 90 minutes allowed for an S1 exam init)
for example the 12th term gives ..
P(12) = (0.5)11 * 0.5 = 0.00024
(if I have done it right)
or in simple terms "not much chance at all"
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| notes
"Distribution" is the way in which the data is arranged. It's characteristics affect the way that we must deal with it.
"infinite"
is a hard word to explain. I could go on forever! It really means that
you could toss a coin until the cows come home and never get a "head".
Unlikely , yes,,but it is true.
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