Advertisements
Advertisements
Question
Let A and B be independent events with P (A) = 0.3 and P (B) = 0.4. Find
- P (A ∩ B)
- P (A ∪ B)
- P (A | B)
- P (B | A)
Advertisements
Solution
It is given that P (A) = 0.3 and P (B) = 0.4
If A and B are independent events, then
(i) P(A ∩ B) = P(A) · P(B)
P(A ∩ B) = 0.3 × 0.4
P(A ∩ B) = 0.12
(ii) P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A ∪ B) = 0.3 + 0.4 - 0.12
P(A ∪ B) = 0.58
(iii) P(A | B) = `(P(A ∩ B))/(P(B))`
P(A | B) = `0.12/0.4`
P(A | B) `= 3/10`
P(A | B) `= 0.3`
(iv) P(B | A) = `(P(A ∩ B))/(P(A))`
P(B | A) = `0.12/0.3`
P(B | A) = 0.4
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?
A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event "number obtained is even" and B be the event "number obtained is red". Find if A and B are independent events.
If `P(A) = 3/5 and P(B) = 1/5` , find P (A ∩ B) if A and B are independent events.
Probability of solving specific problem independently by A and B are `1/2` and `1/3` respectively. If both try to solve the problem independently, find the probability that
- the problem is solved
- exactly one of them solves the problem.
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’
Prove that if E and F are independent events, then the events E and F' are also independent.
The probabilities of solving a specific problem independently by A and B are `1/3` and `1/5` respectively. If both try to solve the problem independently, find the probability that the problem is solved.
One-shot is fired from each of the three guns. Let A, B, and C denote the events that the target is hit by the first, second and third guns respectively. assuming that A, B, and C are independent events and that P(A) = 0.5, P(B) = 0.6, and P(C) = 0.8, then find the probability that at least one hit is registered.
The odds against a certain event are 5: 2 and odds in favour of another independent event are 6: 5. Find the chance that at least one of the events will happen.
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 probability that neither solves the problem?
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.
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:
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:
Find the probability that a year selected will have 53 Wednesdays
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)
If A and B are independent events such that 0 < P(A) < 1 and 0 < P(B) < 1, then which of the following is not correct?
Three events A, B and C are said to be independent if P(A ∩ B ∩ C) = P(A) P(B) P(C).
Refer to Question 1 above. If the die were fair, determine whether or not the events A and B are independent.
The probability that at least one of the two events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.3, evaluate `"P"(bar"A") + "P"(bar"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 – P1) P2
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: P1 + P2 – 2P1P2
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 ______.
If A and B are independent events, then A′ and B′ are also independent
If A and B are mutually exclusive events, then they will be independent also.
Two independent events are always mutually exclusive.
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 independent, then P(exactly one of A, B occurs) = P(A)P(B') + P(B)P(A')
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 spade’
F : ‘the card drawn is an ace’
If A, B are two events such that `1/8 ≤ P(A ∩ B) ≤ 3/8` then
Given two independent events A and B such that P(A) = 0.3, P(B) = 0.6 and P(A' ∩ B') is ______.
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.
The probability of the event A occurring is `1/3` and of the event B occurring is `1/2`. If A and B are independent events, then find the probability of neither A nor B occurring.
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 ______.
Let E and F be two independent events. The probability that both E and F happen is `1/12` and the probability that neither E nor F happens is `1/2`, then a value of `(P(E))/(P(F))` is ______.
