The event ‘E' and the event ‘not E' are called complementary event of the latter event. If E occurs, its compliment is E which does not occur.
Compliment of an event is denoted by E’ or E or E^{c}.For example:
1. When a coin is tossed, getting ‘head’ and getting ‘tail’ are complimentary event of each other.
2. When two coins are tossed, getting ‘atleast one head’ and getting ‘no head’ are complimentary event of each other.
3. When a die is thrown:
● Getting ‘even face’ and ‘odd face’ are complimentary event of each other.
● Getting ‘multiple of 2’ and getting ‘not multiple of 2’ are complement event of the latter event.
● Getting ‘divisible by 3’ and getting ‘not divisible by 3’ are complement event of the latter event.
Sum of the probability of an event ‘E’ and probability of an event ‘not E’ is always 1.Now we will solve the examples on different types of word problems on complementary event.
Problems on complementary event:
1. A bag contains red and what balls. The probability of getting a red ball from the bag of balls is 1/6. What is the probability of not getting a red ball?
Solution:
The probability of getting a red ball from the bag of balls is 1/6.
Therefore, the probability of not getting a red ball
P(ball is not red) = 1 – 1/6 = 5/6
Therefore, the probability of not getting a red ball is 5/6.
2. In a box, contains blue and green marbles. The probability of getting a green marble from the box of marbles is 3/7. What is the probability of getting a blue marble?
Solution:= 1 – 3/7
= 4/7.
Therefore, the probability of getting a blue marble is 4/7.
3. In a cricket tournament Yuvraj Singh hits eight times ‘6’ out of thirty two balls. Calculate the probability that he would not hit a 6?
Solution:
Let P(A) = 32be the total number of events .
Favorable events that is Yuvraj Singh hits a boundary P (B) = 8,
Therefore, P(E) = P (Yuvraj Singh hits a ‘6’)
= P(B)/P(A)
= 8/32 = ¼.
Now, P (not E) = P (Yuvraj Singh did not hit a ‘6’) = 1 – 1/4 = 3/4.
4. In a laptop shop there are 16 defective laptops out of 200 laptops. If one laptop is taken out at random from this laptop shop, what is the probability that it is a non defective laptop?
Solution:
The total number of laptops in laptop shop = 200,
The number of defective laptops = 16,
Let E_{1}be the event of getting a defective laptops andP(A) = The probability of getting a defective laptop
= 16/200
= 0.08
Therefore, the probability of getting a non defective laptop = 1 – P(A) = 1 – 0.08 = 0.92.
5. The probability that it will rain in the evening 0.84. What is the probability that it will not rain in the evening?
Solution:
Let E be the event that it will rain in the evening.
Then, (not E) is the event it will rain in the evening .
Then, P(E) = 0.84
Now, P(E) + P(not E) = 1
⇒ P(not E) = 1  P(E)
⇒ P(not E) = 1  0.84
⇒ P(not E) = 0.16
Therefore, the probability that it will not rain in the evening = P(not E) = 0.16
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