Probability
Category : 9th Class
PROBABILITY
Learning Objectives
Probability
A mathematically measure of uncertainty is known as probability. If there are 'a' elementary events associated with a random experiment and 'b' of them are favourable to event 'E':
\[\therefore \,\,\,\,\,\,P(E)=\frac{b}{a}\] \[\Rightarrow \,\,0\le P\left( E \right)\le 1\]
\[\therefore \,\,\,\,\,\,\,P\,(\overline{E})\,=\frac{a-b}{a}=1-\frac{b}{a}=1-P(E)\]
Experiment
An operation which can produce some well- defined outcomes is called an experiment,
Random Experiment: An experiment in which all possible outcomes are known and exact outcome cannot be predicted is called a random experiment.
Example: Rolling an unbiased dice has all six outcomes (1, 2, 3, 4, 5, 6) known but exact outcome can be predicted.
Outcome: The result of a random experiment is called an outcome,
Sample Space: The set of all possible outcomes of a random experiment is known as sample space.
Example: The sample space in throwing of a dice is the set (1, 2, 3, 4, 5, 6).
Trial: The performance of a random experiment is called a trial.
Example: The tossing of a coin is called trial.
Event
An event is a set of experimental outcomes, or in other words it is a subset of sample space.
Example: On tossing of a dice, let A denotes the event of even number appears on top A: {2, 4, 6}.
Mutually Exclusive Events: Two or more events are said to be mutually exclusive if the occurrence of any one excludes the happening of other in the same experiment. E.g. On tossing of a coin is head occur, then it prevents happing of tail, in the same single experiment.
Exhaustive Events: All possible outcomes of an event are known as exhaustive events. Example:
In a throw of single dice the exhaustive events are six {1, 2, 3, 4, 5, 6}.
Equally Likely Event: Two or more events are said to be equally likely if the chances of their happening are equal.
Example: On throwing an unbiased coin, probability of getting Head and Tail are equal.
Playing Cards
There are four face cards each in number four Ace, King, Queen and Jack.
Black Suit (26) |
Red Suit (26) |
Spade (13) & Club (13) |
Diamond (13) & Heart (13) |
Commonly Asked Questions
(a) \[\frac{1}{2}\] (b) \[\frac{3}{2}\]
(c) \[\frac{1}{3}\] (d) \[\frac{1}{4}\]
(e) None of these
Answer: (a)
Explanation: In this case sample space, S = {H, T}, Event E = {T}
\[\therefore \,\,\,P(E)\,=\frac{n(E)}{n(S)}=\frac{1}{2}\]
(a) \[\frac{1}{4}\] (b) \[\frac{3}{2}\]
(c) \[\frac{1}{3}\] (d) \[\frac{1}{2}\]
Answer (d)
Explanation: S = {1, 2, 3, 4, 5, 6}, Event E = {2, 4, 6} multiple of 2
\[\therefore \,\,\,P(E)\,=\frac{n(E)}{n(S)}=\frac{3}{6}=\frac{1}{2}\]
(a) \[\frac{1}{3}\] (b) \[\frac{2}{5}\]
(c) \[\frac{2}{3}\] (d) \[\frac{1}{6}\]
(e) None of these
Answer: (c)
Explanation: Here Sample space S = {1, 2, 3, 4, 5, 6}, Event E = {1, 2, 3, 4} number- less than or equal to 4.
\[\therefore \,\,\,P(E)\,=\frac{n(E)}{n(S)}=\frac{4}{6}=\frac{2}{3}\]
(a) \[\frac{2}{5}\] (b) \[\frac{2}{7}\]
(c) \[\frac{1}{7}\] (d) \[\frac{2}{4}\]
(e) None of these
Answer: (b)
Explanation: A leap year has 366 days, out of which there are 52 weeks and 2 more days.
2 more days can be (Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday),
(Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday), (Saturday, Sunday)
= n(S) = 7
So, (Sunday, Monday) and (Saturday, Sunday) = n (E) = 2, therefore chances that a leap year selected randomly will have 53 Sundays:
\[\therefore \,\,\,P(E)\,=\frac{n(E)}{n(S)}=\frac{2}{7}\]
(a) \[\frac{1}{7}\] (b) \[\frac{2}{7}\]
(c) \[\frac{2}{8}\] (d) \[\frac{2}{6}\]
(e) None of these
Answer: (a)
Explanation: A normal year has 365 days, out of which there are 52 weeks and 1 more day
So, extra day can be Sunday, Monday, Tuesday, Wednesday, Thursday Friday, Saturday
So, n(S) = 7, n(E) = 1
\[\therefore \,\,\,P(E)\,=\frac{n(E)}{n(S)}=\frac{1}{7}\]
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