Probability - Important Concepts and Notes

BASICS : 

We perform an experiment which can have a number of different outcomes. The sample space is the set of all possible outcomes of the experiment. We usually call it S.
For example, if I plant ten bean seeds and count the number that germinate, the sample space is

S = {0,1,2,3,4,5,6,7,8,9,10}.

If I toss a coin three times and record the result, the sample space is

S = {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT},

where (for example) HTH means ‘heads on the first toss, then tails, then heads again.

An event is a subset of S. We can specify an event by listing all the outcomes that make it up. In the above example, let A be the event ‘more heads than tails’ and B the event ‘heads on last throw’. Then

A = {HHH,HHT,HTH,THH}
B = {HHH,HTH,THH,TTH}

Probability = Favorable number of cases/Total Number of cases

Hence if asked what is the probability of event A in above example , then

Probability = n(A)/n(S) = 4/8 = 0.5

We can build new events from old ones:
• A∪B (read ‘A union B’) consists of all the outcomes in A or in B (or both!)
• A∩B (read ‘A intersection B’) consists of all the outcomes in both A and B;
• A\B (read ‘A minus B’) consists of all the outcomes in A but not in B;
• A' (read ‘A complement’) consists of all outcomes not in A (that is, S \A);
• Ø (read ‘empty set’) for the event which doesn’t contain any outcomes

Independent Events : 

Two events A and B are said to be independent if

P(A∩B) = P(A)·P(B)

Mutually Exclusive Events : 

Two events A and B are said to be mutually exclusive if

P(A∩B) = 0

Collectively Exhaustive Events : 

Two events A and B are said to be collectively exhaustive if

P(AUB) = 1

CONDITIONAL PROBABILITY :

Let E be an event with non-zero probability, and let A be any event. The conditional probability of A given E is defined as

P(A | E) = P(A∩E) /P(E)

Example : A random car is chosen among all those passing through Trafalgar Square on a certain day. The probability that the car is yellow is 3/100: the probability that the driver is blonde is 1/5; and the probability that the car is yellow and the driver is blonde is 1/50. Find the conditional probability that the driver is blonde given that the car is yellow.

Solution: If Y is the event ‘the car is yellow’ and B the event ‘the driver is blonde’, then we are given that P(Y) = 0.03, P(B) = 0.2, and P(Y ∩B) = 0.02. So
P(B | Y) = P(B∩Y) / P(Y) = 0.02 /0.03 = 0.667

BAYES' Theorem :

There is a very big difference between P(A | B) and P(B | A).

Let A and B be events with non-zero probability. Then

So this was about basics of probability and conditional probability . Practice questions now from practice section in the app.

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