4. Probability of an event

Probability of an event

The probability of an event is the proportion (relative frequency) of times that the event is expected to occur when an experiment is repeated a large number of times under identical conditions.

Where,

• P(A) is the probability of an event “A”
• n(A) is the number of favourable outcomes
• n(S) is the total number of events in the sample space

What is the probability to get a 6 when you roll a die?

A die has 6 sides, 1 side contain the number 6 that give us 1 wanted outcome in 6 possible outcomes.

Example:

Let us consider another experiment of ‘tossing a coin “twice”

The sample space of this experiment is S = {HH, HT, TH, TT}

P(HH) =1/4  , P(HT) = 1 / 7 , P(TH) = 2 / 7 ,  P(TT) = 9 / 28

The probability of the event E: ‘Both the tosses yield the same result’.

Here E = {HH, TT}

Now P(E) = S P(wi), for all wi Î E

= P(HH) + P(TT) = 1/4  + 9 / 28  = 4 /7

For the event F: ‘exactly two heads’, we have F = {HH}

P(F) = P(HH) = ¼

Probabilities of equally likely outcomes: Let a sample space of an experiment be

S = {w1, w2,..., wn}.

Let all the outcomes are equally likely to occur, i.e., the chance of occurrence of each

simple event must be same.

i.e. P(wi) = p, for all wi Î S where 0 £ p £ 1

Since  i.e., P(w1)+ P(w2)+ P(w3) + P(w4) + P(w5)+.....+ P(wn) = p + p + ... + p (n times) = 1

Let S be a sample space and E be an event, such that n(S) = n and n(E) = m. If each outcome is equally likely, then it follows that

Basic Probability Formulas

Let A and B are two events. The probability formulas are listed below: