Advertisements
Advertisements
Question
The probability that a student X solves a problem in dynamics is `2/5` and the probability that student Y solves the same problem is `1/4`. What is the probability that
- the problem is not solved
- the problem is solved
- the problem is solved exactly by one of them
Advertisements
Solution
Let event A: Student X solves the problem in dynamics,
event B: Student Y solves the problem in dynamics.
∴ P(A) = `2/5`, P(B) = `1/4`
∴ P(A') = 1 – P(A) = `1 - 2/5 = 3/5`
P(B') = 1 – P(B) = `1 - 1/4 = 3/4`
A and B are independent events,
A' and B' are also independent events
(i) Let event C: Problem is not solved.
∴ P(C) = P(A' ∩ B')
= P(A') · P(B')
= `3/5 xx 3/4`
= `9/20`
(ii) Let event D: Problem is solved.
Problem can be solved if at least one of the two students solves the problem.
∴ P(C) = P(at least one student solves the problem)
= 1 - P (no student solves the problem)
= 1 - P(A' ∩ B')
= 1 - P (A') · P (B')
`= 1 - 3/5 xx 3/4`
`= 1 - 9/20`
`= 11/20`
(iii) Let event E: The problem is solved exactly by one of them.
∴ P(E) = P(A' ∩ B) ∪ P(A ∩ B')
= P(A') · P (B) + · P (A) · P (B')
`= (3/5 xx 1/4) + (2/5 xx 3/4)`
`= 3/20 + 6/20`
`= 9/20`
APPEARS IN
RELATED QUESTIONS
A bag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white?
If A and B are two independent events such that `P(barA∩ B) =2/15 and P(A ∩ barB) = 1/6`, then find P(A) and P(B).
If `P(A) = 3/5 and P(B) = 1/5` , find P (A ∩ B) if A and B are independent events.
If A and B are two events such that `P(A) = 1/4, P(B) = 1/2 and P(A ∩ B) = 1/8`, find P (not A and not B).
Prove that if E and F are independent events, then the events E and F' are also independent.
A problem in statistics is given to three students A, B, and C. Their chances of solving the problem are `1/3`, `1/4`, and `1/5` respectively. If all of them try independently, what is the probability that, problem is not solved
An urn contains four tickets marked with numbers 112, 121, 122, 222 and one ticket is drawn at random. Let Ai (i = 1, 2, 3) be the event that ith digit of the number of the ticket drawn is 1. Discuss the independence of the events A1, A2, and A3.
The odds against a husband who is 55 years old living till he is 75 is 8: 5 and it is 4: 3 against his wife who is now 48, living till she is 68. Find the probability that at least one of them will be alive 20 years hence.
The odds against student X solving a business statistics problem are 8: 6 and odds in favour of student Y solving the same problem are 14: 16 What is the chance that the problem will be solved, if they try independently?
A bag contains 3 yellow and 5 brown balls. Another bag contains 4 yellow and 6 brown balls. If one ball is drawn from each bag, what is the probability that, the balls are of different color?
Bag A contains 3 red and 2 white balls and bag B contains 2 red and 5 white balls. A bag is selected at random, a ball is drawn and put into the other bag, and then a ball is drawn from that bag. Find the probability that both the balls drawn are of same color
A family has two children. Find the probability that both the children are girls, given that atleast one of them is a girl.
Select the correct option from the given alternatives :
The odds against an event are 5:3 and the odds in favour of another independent event are 7:5. The probability that at least one of the two events will occur is
Solve the following:
If P(A ∩ B) = `1/2`, P(B ∩ C) = `1/3`, P(C ∩ A) = `1/6` then find P(A), P(B) and P(C), If A,B,C are independent events.
Solve the following:
If P(A) = `"P"("A"/"B") = 1/5, "P"("B"/"A") = 1/3` the find `"P"("B'"/"A'")`
Solve the following:
Find the probability that a year selected will have 53 Wednesdays
The probability of simultaneous occurrence of at least one of two events A and B is p. If the probability that exactly one of A, B occurs is q, then prove that P(A′) + P(B′) = 2 – 2p + q.
Two dice are thrown together. Let A be the event ‘getting 6 on the first die’ and B be the event ‘getting 2 on the second die’. Are the events A and B independent?
If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one of A, B) = `5/9`, then p = ______.
Let A and B be two independent events. Then P(A ∩ B) = P(A) + P(B)
A and B are two events such that P(A) = `1/2`, P(B) = `1/3` and P(A ∩ B) = `1/4`. Find: `"P"("A"/"B")`
A and B are two events such that P(A) = `1/2`, P(B) = `1/3` and P(A ∩ B) = `1/4`. Find: `"P"("A'"/"B")`
Three events A, B and C have probabilities `2/5, 1/3` and `1/2`, , respectively. Given that P(A ∩ C) = `1/5` and P(B ∩ C) = `1/4`, find the values of P(C|B) and P(A' ∩ C').
Let E1 and E2 be two independent events such that P(E1) = P1 and P(E2) = P2. Describe in words of the events whose probabilities are: 1 – (1 – P1)(1 – P2)
If A and B are two events and A ≠ Φ, B ≠ Φ, then ______.
If two events are independent, then ______.
Let P(A) > 0 and P(B) > 0. Then A and B can be both mutually exclusive and independent.
If A and B are two independent events then P(A and B) = P(A).P(B).
If A and B′ are independent events, then P(A' ∪ B) = 1 – P (A) P(B')
If A and B are two events such that P(A) > 0 and P(A) + P(B) >1, then P(B|A) ≥ `1 - ("P"("B'"))/("P"("A"))`
If A, B and C are three independent events such that P(A) = P(B) = P(C) = p, then P(At least two of A, B, C occur) = 3p2 – 2p3
If A and B are two events such that P(A|B) = p, P(A) = p, P(B) = `1/3` and P(A ∪ B) = `5/9`, then p = ______.
Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.
Given two independent events, if the probability that exactly one of them occurs is `26/49` and the probability that none of them occurs is `15/49`, then the probability of more probable of the two events is ______.
Five fair coins are tossed simultaneously. The probability of the events that at least one head comes up is ______.
Two players A and B are alternately throwing a coin and a die together. A player who first throws head and 6 wins the game. If A starts the game, then the probability that B wins the game is ______.
