Notes - Probability

**Category : **11th Class

**Learning Objectives**

- Probability
- Playing Cards

**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';

- Then the probability of occurrence of event E is denoted by P(e).

\[\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,P(E)=\frac{b}{a}\,\,\,\,\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\,\,\,\,0\le P(E)\le 1\]

- The probability of non-occurrence of event E denoted by P(e) and is defined as \[\frac{a-b}{a}.\]

\[\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,P(\overline{E})=\frac{a-b}{a}=1-\frac{b}{a}=1-P(E)\]

- \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,P(E)+P(\overline{E})=1\]

**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 unbiassed coin, probability of getting Head and Tail are equal.

**Playing Cards**

- Total number of card are 52.
- There are 13 cards of each suit named Diamond, Hearts, Clubs and Spades.
- Out of which Hearts and diamonds are red cards.
- Spades and Clubs are black cards.
- There are four face cards each in number four Ace, king, Queen and jack.

Black Suit (26) |
Red Suit (26) |

Spades (13) & Club (13) |
Diamond (13) & Heart (13) |

- Each Spade, Club, Diamond, Heart has 9 digit cards 2, 3, 4, 5, 6, 7, 8, 9, and 10.
- There are 4 Honour cards each Spade, Club, Diamond, Heart Contains 4 numbers of Honours cards Ace, king, Queen and jack.

**Commonly Asked Questions**

**In a through of a coin find the probability of getting a tail.**

(a) \[\frac{1}{2}\] (b) \[\frac{3}{2}\]

(c) \[\frac{1}{3}\] (d) \[\frac{1}{4}\]

(e) None of these

**Ans. (a)**

**Explanation:** In this case sample space, S = {H, T}, Event E = {T}

\[\therefore \,\,\,P(E)=\frac{n(E)}{n(S)}=\frac{1}{2}\]

**An unbiased die is tossed. Find the probability of getting a multiple of 2.**

(a) \[\frac{1}{4}\] (b) \[\frac{3}{2}\]

(c) \[\frac{1}{3}\] (d) \[\frac{1}{4}\]

(e) None of these

**Ans. (d)**

**Explanation:** Here Sample space S = {1, 2, 3, 4, 5, 6}, Event E = {2, 4, 6} multiple of 2

\[\therefore \,\,\,P(E)=\frac{n(E)}{n(S)}=\frac{3}{6}=\frac{1}{2}\]

**An unbiased die is tossed. Find the probability of getting a number less than or equal to 4.**

(a) \[\frac{1}{3}\] (b) \[\frac{2}{5}\]

(c) \[\frac{2}{3}\] (d) \[\frac{1}{6}\]

(e) None of these

**Ans. (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.

\[\therefore \,\,\,P(E)=\frac{n(E)}{n(S)}=\frac{4}{6}=\frac{2}{3}\]

**What is the chance that a leap year selected randomly will have 53 Sundays?**

(a) \[\frac{2}{5}\] (b) \[\frac{2}{7}\]

(c) \[\frac{1}{7}\] (d) \[\frac{2}{4}\]

(e) None of these

**Ans. (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.

\[\therefore \,\,\,P(E)=\frac{n(E)}{n(S)}=\frac{2}{7}\]

**What is the chance that a normal year selected randomly will have 53 Sundays?**

(a) \[\frac{1}{7}\] (b) \[\frac{2}{7}\]

(c) \[\frac{2}{8}\] (d) \[\frac{2}{6}\]

(e) None of these

**Ans. (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

\[\therefore \,\,\,P(E)=\frac{n(E)}{n(S)}=\frac{1}{7}\].

*play_arrow*Notes - Probability

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