Let A & B two events defined on a sample space. Given P A = 0.4 ; P B = 0.80 and P A ¯ B ¯ = 0.10 . Then find : i P A ¯ ∪ B ii P A ¯ ∩ B ∪ A ∩ B ¯

Given: P A = 0.4 ⇒ P A ¯ = 1 - 0 . 4 = 0 . 6 P B = 0.80 ⇒ P B ¯ = 1 = 0 . 80 = 0 . 20 And, P A ¯ B ¯ = P A ¯ ∩ B ¯ P B ¯ = 0.1 ⇒ P A ∪ B ¯ = 0.02 We know that, P A ¯ ∪ B = P A ¯ + P B − P A ¯ ∩ B ⇒ P A ¯ ∪ B = P A ¯ + P B − P B − P A ∩ B ⇒ P A ¯ ∪ B = P A ¯ + P A ∩ B . . . . 1 Now, P A ∪ B = 1 - P A ∪ B ¯ = 0.98 And, P A ∪ B = P A + P B - P A ∩ B = 0 . 98 ⇒ P A ∩ B = 0.4 + 0.8 − 0.98 = 0.22 . . . . 2 Hence, by 1 P A ¯ ∪ B = 0.6 + 0 . 22 = 0 . 82 And, P A ¯ ∩ B ∪ A ∩ B ¯ = P A ¯ ∪ B + P A ∩ B ¯ = P A ¯ ∪ B + P A - P A ∩ B = 0 . 82 + 0 . 4 - 0 . 22 = 1

A is a set containing n elements. A subset P of A is chosen at random. The set A is reconstructed by replacing the elements of P . A subset Q of A is again chosen at random. Find the probability that P and Q have no common elements.

Given: set A having n elements. Subset P of set A is chosen at random. After the reconstruction of the set A by replacing the element of P , the subset Q is again chosen at random Here set A having n elements Total number of subset = 2 n Number of ways by which we can choose a subset P = C 1 2 n = 2 n Similarly, the set Q can be chosen in 2 n ways. Now P and Q can be chosen = C 1 2 n × C 1 2 n = ( 2 n ) ( 2 n ) = 4 n ways (Total outcomes) Now P and Q should be disjoint set ⇒ P ∩ Q = ∅ Let Set P contains i elements out of n So the number of ways = C i n Now remaining elements = n - i ⇒ Q can be chosen out of these n - i elements = n - i C 0 + n - i C 1 + . . . . . + n - i C n - i = 2 n - i ∴ P and Q can be chosen in n C i 2 n - i ways but i can vary from 0 to n ⇒ ∑ i = 0 n n C i 2 n - i = ( 1 + 2 ) n = 3 n Hence, required probability = 3 n 4 n = 3 4 n

HARD

JEE Main/Advance

IMPORTANT

Earn 100

The probabilities that three men hit a target are, respectively, $0.3, 0.5$ and $0.4$ . Each fires once at the target. (As usual, assume that the three events that each hits the target are independent)

$(a)$ Find the probability that they all : (i) hit the target ; (ii) miss the target

$(b)$ Find the probability that the target is hit: (i) atleast once, (ii) exactly once.

$(c)$ If only one hits the target, what is the probability that is was the first man?

Important Questions on Probability

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JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 11 - Probability>EXERCISE-4>Q 6

Let $(i) [1 - P (\bar{A} \cap \bar{B} \cap \bar{C})] = 0.79$ & $B$ two events defined on a sample space. Given $P (A) = 0.4; P (B) = 0.80$ and $P (\frac{\bar{A}}{\bar{B}}) = 0.10$ . Then find :

$(i) P (\bar{A} \cup B)$

$(ii) P [(\bar{A} \cap B) \cup (A \cap \bar{B})]$

HARD

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 11 - Probability>EXERCISE-4>Q 7

Three shots are fired at a target in succession. The probabilities of a hit in the first shot is

\frac{1}{2}

, in the second

\frac{2}{3}

and in the third shot is

\frac{3}{4}

. In exactly one hit, the probability of destroying the target is

\frac{1}{3}

and in the case of exactly two hits

\frac{7}{11}

and in the case of three hits is

1.0

. Find the probability of destroying the target in three shots.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 11 - Probability>EXERCISE-4>Q 8

A certain drug, manufactured by a Company is tested chemically for its toxic nature. Let the event "THE DRUG IS TOXIC" be denoted by

H

& the event "THE CHEMICAL TEST REVEALS THAT THE DRUG IS TOXIC" be denoted by

S

. Let

P (H) = a, P (\frac{S}{H}) = P (\frac{\bar{S}}{\bar{H}}) = 1 - a

. Then show that the probability that the drug is not toxic given that the chemical test reveals that it is toxic is free from '

a

HARD

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 11 - Probability>EXERCISE-4>Q 9

A plane is landing. If the weather is favourable. The pilot landing the plane can see the runway. In this case the probability of a safe landing is

p_{1}

. If there is a low cloud ceiling, the pilot has to make a blind landing by instruments. The reliability (the probability of failure free functioning) of the instruments needed for a blind landing is

P

. If the blind landing instruments function normally, the plane makes a safe landing with the same probability

p_{1}

as in the case of a visual landing. If the blind instruments fail, then the pilot may make a safe landing with probability

p_{2} < p_{1}

. Compute the probability of a safe landing if it is known that in

K

percent of the cases there is a low cloud ceiling. Also find the probability that the pilot used the blind landing instruments, if the plane landed safely.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 11 - Probability>EXERCISE-4>Q 10

A

is a set containing

n

elements.

A

subset

P

A

is chosen at random. The set

A

is reconstructed by replacing the elements of

P . A

subset

Q

A

is again chosen at random. Find the probability that

P

and

Q

have no common elements.

MEDIUM

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 11 - Probability>EXERCISE-4>Q 11

In a multiple choice question there are five alternative answers of which one or more than one is correct. A candidate will get marks on the question only if he ticks the correct answers. The candidate ticks the answers at random. If the probability of the candidate getting marks on the question is to be greater than or equal to

\frac{1}{3}

, find the least number of chances he should be allowed.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 11 - Probability>EXERCISE-4>Q 12

The ratio of the number of trucks along a highway, on which a petrol pump is located, to the number of cars running along the same highway is

3 : 2

. It is known that an average of one truck in thirty trucks and two cars in fifty cars stop at the petrol pump to be filled up with the fuel. If a vehicle stops at the petrol pump to be filled up with the fuel, find the probability that it is a car.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 11 - Probability>EXERCISE-4>Q 13

There are

6

red balls &

8

green balls in a bag.

5

balls are drawn out at random & placed in a red box; the remaining

9

balls are put in a green box. What is the probability that the number of red balls in the green box plus the number of green balls in the red box is not a prime number?

Important Questions on Probability

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