| In life, things are not always certain. Consider the following situations: |
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- A candidate appearing for an interview for a job may or may not get the job
- It may or may not snow today.
- If a coin is tossed you might get a head or a tail (or neither if the coin falls on its edge).
- When a dice is thrown your chance of getting a 6 may or may not occur, since it is equally likely that a 1, 2, 3, 4, or 5 may turn up.
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These situations have no definite answer. They involve an element of uncertainty. The 'Probability Theory' is designed to estimate the degree of uncertainty regarding the happening of a given phenomenon.
Probability is used in various situations in physical, biological, and social sciences.
Suppose you toss a dice once, what are the possible outcomes? A dice, obviously can fall with any of its faces uppermost. The number on each face is a possible outcome. If the dice is well-balanced, it is likely to show a 2, or a 1, 3, 4, 5, 6.
Since there are 6 equally likely outcomes 1, 2, 3, 4, 5 or 6 in a single throw of a dice and there is only one way of getting a particular outcome say ‘ 6 ', therefore the chance of getting 6 is one in six. Or the probability of getting a 6 is 1/6. We write this as P (6) = 1/6
Similarly if you toss a coin it can show a head (H) or a tail (T). So there are only two equally likely outcomes. The probability of getting a tail is one in two or 1/2. |
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The outcome is also called an event (E). We write the probability of an event as P (E) and define it as
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| In the case of tossing a dice, the total number of outcomes in the set {1, 2, 3, 4, 5, 6} is 6. If we want a 6 then we just have 1 favorable outcome as there's only one outcome of 6 on a dice. |
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| The set {1, 2, 3, 4, 5, 6} is called a Sample space and each Outcome is called a Sample point. Tossing a coin or a dice is called a Random |
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| Experiment. |
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| What is the probability of getting an '8' if a dice is tossed once? Since none of the faces is marked by an 8, getting an eight is impossible. Such an event is called an impossible event and we have |
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| The Probability of an impossible event is zero. |
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The probability of getting a 1, 2, 3, 4, 5 or 6 when a dice is tossed i.e. a number less than 7 is certain to happen or P (< 7) = 6/6 = 1.
This is called a sure event.
We know that the probability of getting a number 6 in the throw of a dice is 1/6. What is the probability of getting a number other than 6? The numbers are 1, 2, 3, 4 or 5 or 5 favorable outcomes. |
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We can write P (other than 6) as P (not 6)
Then |
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We thus have the formula for any event E
P (E) + P ( Ē ) = 1
and P (E) = 1-P( Ē )
P (Ē) = 1- P (E)
P ( Ē) indicates P(not E).
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| Example 19: |
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A dice is thrown once. What is the probability of getting
- A number 3 or 4?
- An odd number?
- A prime number?
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| Solution: |
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i. |
Number of favorable outcomes getting a 3 or 4 = 2
Total number of outcomes = 6
Required probability: P(E)
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ii. |
Number of odd numbers = 3, since the only odd numbers are 1, 3, and 5
Number of favorable outcomes = 3
Total number of outcomes = 6
Probability of event = P(E)
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iii. |
Number of prime number = 3, since the only prime numbers are (2, 3, 5)
Number of favorable outcomes = 3
Total number of outcomes = 6
Probability of the event = P(E)
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| Example 20: |
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| If one card is drawn from a well shuffled deck of 52 cards, find the probability that the card is
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- a diamond
- an ace
- a black card
- not a diamond
- not an ace
- not a black card
- a club or a heart
- a club and a king
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Solution: |
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i. |
In a pack of 52 cards there are 13 diamond cards
Number of favorable outcomes = 13.
Total number of outcomes = 52
Probability of the event = P(E)
(getting diamond)
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ii. |
Number of aces in a deck of 52 cards is 4
Number of favorable outcomes = 4
Total number of outcomes = 52
Probability of the event = P(E)
(getting an ace)
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iii. |
The number of black cards in a deck of 52 cards is 26
13 of clubs
13 of spades
Number of favorable outcomes = 26
Total number of outcomes = 52
Probability of the event = P(E)
(getting a black card)
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iv. |
Since the probability P (E) of getting a diamond is 1/4
The probability of not getting a diamond
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Probability of getting an ace = P (E) = 1/13
Probability of not getting an ace = 1-P (E)
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vi. |
Probability of getting a black card = P (E) = 1/2
Probability of not getting a black card = P (Ē) = 1 – P (E)
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vii. |
Probability of getting a club or a heart
Number of club cards in a deck = 13
Number of heart cards in a deck = 13
Number of favorable outcomes = 13 + 13 = 26
Total number of outcomes = 52
Probability of the event = P(E)
(getting a club or a heart)
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viii. |
A club and a king
Since there is only one king of clubs
Number of favorable outcomes = 1
Total number of outcomes = 52
Probability of the event = P(E)
(getting a club or a king)
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| Try these problems: |
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- A dice is thrown twice. Find the probability of getting a sum of six in these throws.
- A coin is tossed thrice, write its sample space.
- From a well shuffled deck of cards find the probability of
a. Getting '2' of hearts.
b. Getting a king or a queen or a jack.
c. Not getting an ace.
d. Getting a red card
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| Answer: |
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1. A dice is thrown twice.
Sample space S = {(1,1), (1, 2),(1,3), (1,4),(1,5), (1,6)
(2,1), (2, 2),(2,3), (2, 4),(2,5), (2, 6)
(3,1), (3, 2),(3,3), (3, 4),(3,5), (3, 6)
(4,1), (4, 2),(4,3), (4, 4),(4,5), (4, 6)
(5,1), (5, 2),(5,3), (5, 4),(5,5), (5, 6)
(6,1), (6, 2),(6,3), (6, 4),(6,5), (6, 6)}
Total number of outcomes= 36 = n(S)
A = event of getting a sum of six
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2. A coin is tossed thrice
Sample space S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT}
Total number of outcomes = 23 = 2 * 2 * 2 = 8
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3. Total number of outcomes = Total number of cards
= n(S)
n(S) = 52
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a. |
A = event of getting ' 2 ' of hearts
n(A) =1 = number of favorable outcomes
Probability of getting a ' 2 ' of hearts is  |
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b. |
A = event of getting a king
n(A) = number of favorable outcomes = 4
B = event of getting a queen
n(B) = number of favorable outcomes = 4
C = event of getting a jack
n(c) = number of favorable outcomes = 4
Since getting a king or a queen or a jack are mutually exclusive events
P(A or B or C) = P (A ∪ B ∪ C) = P(A) + P (B) + P (C)
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c. |
A = event of getting an ace
n(A) = number of favorable outcomes = 4
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d. |
R = event of getting a red card
n(R) = number of favorable outcomes
= 26 (13 diamonds + 13 hearts)
Probability of getting a red card is  |
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