Advertisements
Advertisements
प्रश्न
A problem in Mathematics is given to three students whose chances of solving it are `1/2, 1/3, 1/4` respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is ______.
पर्याय
`1/4`
`1/3`
`1/2`
`3/4`
Advertisements
उत्तर १
A problem in Mathematics is given to three students whose chances of solving it are `1/2, 1/3, 1/4` respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is `underlinebb(3/4)`.
Explanation:
Let A, B, C be the respective events of solving the problem.
Then, P(A) = `1/2`, P(B) = `1/3` and P(C) = `1/4`.
Here, A, B, C are independent events.
Problem is solved if at least one of them solves the problem.
Required probability is
= P(A ∪ B ∪ C)
= `1 - P(overlineA)P(overlineB)P(overlineC)`
= `1 - (1 - 1/2)(1 - 1/3)(1 - 1/4)`
= `1 - 1/4`
= `3/4`.
उत्तर २
A problem in Mathematics is given to three students whose chances of solving it are `1/2, 1/3, 1/4` respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is `underlinebb(3/4)`.
Explanation:
The problem will be solved if one or more of them can solve the problem.
The probability is
`P(Aoverline(BC)) + P(overlineABoverlineC) + P(overline(AB)C) + P(ABoverlineC) + P(AoverlineBC) + P(overlineABC) + P(ABC)`
= `1/2. 2/3. 3/4 + 1/2. 1/3. 3/4 + 1/2 . 2/3. 1/4 + 1/2. 1/3. 3/4 + 1/2. 2/3. 1/4 + 1/2. 1/3. 1/4 + 1/2. 1/3. 1/4`
= `3/4`.
उत्तर ३
A problem in Mathematics is given to three students whose chances of solving it are `1/2, 1/3, 1/4` respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is `underlinebb(3/4)`.
Explanation:
Let us think quantitively.
Let us assume that there are 100 questions given to A.
A solves `1/2 xx 100` = 50 questions then remaining 50 questions is given to B and B solves `50 xx 1/3` = 16.67 questions.
Remaining `50 xx 2/3` questions is given to C and C solves `50 xx 2/3 xx 1/4` = 8.33 questions.
Therefore, number of questions solved is 50 + 16.67 + 8.33 = 75.
So, required probability is `75/100 = 3/4`.
APPEARS IN
संबंधित प्रश्न
A speaks truth in 60% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? In the cases of contradiction do you think, the statement of B will carry more weight as he speaks truth in more number of cases than A?
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?
A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are independent events or not.
Given two independent events A and B such that P (A) = 0.3, P (B) = 0.6. Find
- P (A and B)
- P(A and not B)
- P(A or B)
- P(neither A nor B)
One card is drawn at random from a well-shuffled deck of 52 cards. In which of the following case is the events E and F independent?
E : ‘the card drawn is a king or queen’
F : ‘the card drawn is a queen or jack’
Two events, A and B, will be independent if ______.
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 the couple will be alive 20 years hence.
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
Two hundred patients who had either Eye surgery or Throat surgery were asked whether they were satisfied or unsatisfied regarding the result of their surgery
The follwoing table summarizes their response:
| Surgery | Satisfied | Unsatisfied | Total |
| Throat | 70 | 25 | 95 |
| Eye | 90 | 15 | 105 |
| Total | 160 | 40 | 200 |
If one person from the 200 patients is selected at random, determine the probability that the person was satisfied given that the person had Throat surgery.
Two hundred patients who had either Eye surgery or Throat surgery were asked whether they were satisfied or unsatisfied regarding the result of their surgery.
The following table summarizes their response:
| Surgery | Satisfied | Unsatisfied | Total |
| Throat | 70 | 25 | 95 |
| Eye | 90 | 15 | 105 |
| Total | 160 | 40 | 200 |
If one person from the 200 patients is selected at random, determine the probability the person had Throat surgery given that the person was unsatisfied.
A bag contains 3 red and 5 white balls. Two balls are drawn at random one after the other without replacement. Find the probability that both the balls are white.
Solution: Let,
A : First ball drawn is white
B : second ball drawn in white.
P(A) = `square/square`
After drawing the first ball, without replacing it into the bag a second ball is drawn from the remaining `square` balls.
∴ P(B/A) = `square/square`
∴ P(Both balls are white) = P(A ∩ B)
`= "P"(square) * "P"(square)`
`= square * square`
= `square`
A family has two children. Find the probability that both the children are girls, given that atleast one of them is a girl.
Solve the following:
If P(A) = `"P"("A"/"B") = 1/5, "P"("B"/"A") = 1/3` the find `"P"("B'"/"A'")`
Solve the following:
Let A and B be independent events with P(A) = `1/4`, and P(A ∪ B) = 2P(B) – P(A). Find `"P"("A"/"B")`
Solve the following:
A and B throw a die alternatively till one of them gets a 3 and wins the game. Find the respective probabilities of winning. (Assuming A begins the game)
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 such events that P(A) > 0 and P(B) ≠ 1, then P(A′|B′) equals ______.
If the events A and B are independent, then P(A ∩ B) is equal to ______.
Let P(A) > 0 and P(B) > 0. Then A and B can be both mutually exclusive and independent.
If A and B are mutually exclusive events, then they will be independent also.
Let A and B be two events. If P(A | B) = P(A), then A is ______ of B.
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.
Let A and B be independent events P(A) = 0.3 and P(B) = 0.4. Find P(A ∩ B)
Events A and Bare such that P(A) = `1/2`, P(B) = `7/12` and `P(barA ∪ barB) = 1/4`. Find whether the events A and B are independent or not.
A die is thrown once. If F is the event that the number appearing is even, which set represents F?
For \[E=\{3,6\}\] when a die is thrown once, what is \[P(E)\]?
For \[E\cap F=\{6\}\] when a die is thrown once, what is \[P(E\cap F)\]?
