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**D. THE PROBABILITY THAT AN EVENT DOES NOT HAPPEN**

If, in a possibility space of n equally likely occurrences, the number of times an event A occurs is r, there are n - r occasions when A does not happen. 'The event A does not happen’ is denoted by (and is read as ‘not A’)

This relationship is most useful in the ‘at least one’ type of problem, as is illustrated below.**Example :** If four cards are drawn at random from a pack of fifty-two playing cards find the probability that at least one of them is an ace.

If A is a combination of four cards containing at least one ace. (i.e. either one ace, or two aces, or three aces or four aces) then is a combination of four cards containing no aces.

Now P ( ) =

Using P (A ) + P ( ) = 1 we have P(A) = 1 - P ( ) = 1 - 0.72 = 0.28

**E. VENN DIAGRAMS**

A diagram used to illustrate relationships between sets. Commonly, a rectangle represents the universal set and a circle within it represents a given set (all members of the given set are represented by points within the circle). A subset is represented by a circle within a circle and union and intersection are indicated by overlapping circles. Let S is the sample space of an experiment and A, B, C are three events corresponding to it

**Example : Let us now conduct an experiment of tossing a pair of dice. Two events defined on the experiment are **

** **

A : getting a doublet {11, 22, 33, 44, 55, 66}

B : getting total score of 10 or more {64, 46, 55, 56, 65, 66}

**F****. ADDITION THEOREM**

A ∪ B = A + B = A or B denotes occurrence of at least A or B.

For 2 events A & B : P (A ∪ B) = P (A) + P (B) - P (A ∩ B)

**Note : **

(a) P( A ∪ B) P(A + B) P(A or B) P (occurrence of atleast one A or B) | P(A) + P(B) – P( A ∩ B) (This is known as generalized addition theorem) P(A) + P(B ∩ ) = P(B) + P(A ∩ ) P(A ∩ B ) + P(A ∩ B) + P(B ∩ ) |

(b) P(A ∪ B) = P(A . ) + P( . B) + P(A . B) = 1-P( . ) | |

(c) Opposite of “atleast A or B” is neither A nor B i.e. |

**Note that **P(A + B) + P( ∩ ) = 1.

**(d)** If A & B are mutually exclusive then P( A ∪ B) = P( A ) + P(B) .**(e) **For any two events A & B, P (exactly one of A, B occurs) =**(f) De Morgan’s Law : **If A & B are two subsets of a universal set U, then

(i) (ii)

**(g) **A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) & A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

**Ex.10 A bag contains 6 white, 5 black and 4 red balls. Find the probability of getting either a white or a black ball in a single draw. Sol. **Let A = event that we get a white ball, B = event that we get a black ball So, the events are mutually exclusive So P( A + B) = P( A ) + P(B) =

**Ex.11 Three numbers are chosen at random without replacement from 1, 2, 3, ......, 10. The probability that the minimum of the chosen numbers is 4 or their maximum is 8, is **

**Sol. **The probability of 4 being the minimum number =

(because, after selecting 4 any two can be selected from 5, 6, 7, 8, 9, 10).

The probability of 8 being the maximum number =

The probability of 4 being the minimum number and 8 being the maximum number =

∴ the required probability = P( A ∪ B) = P( A ) + P(B) - P( A ∩ B) =

**Ex.12 A pair of dice is rolled together till a sum of either 5 or 7 is obtained. Find the probability that 5 comes before 7. Sol.** Let E

Let E

∴ the probability of getting neither 5 nor 7 =

The event of getting 5 before

∴ the probability of getting 5 before

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