Advertisements
Advertisements
Question
If P(A ∪ B) = P(A ∩ B) for any two events A and B, then
Options
P(A) = P(B)
P(A) > P(B)
P(A) < P(B)
None of these
Solution
We know that
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
⇒ P(A ∩ B) = P(A) + P(B) − P(A ∩ B) [P(A ∪ B) = P(A ∩ B)]
⇒ P(A) − P(A ∩ B) + P(B) − P(A ∩ B) = 0 .....(1)
But,
P(A) − P(A ∩ B) ≥ 0
P(B) − P(A ∩ B) ≥ 0
⇒ P(A) − P(A ∩ B) + P(B) − P(A ∩ B) ≥ 0 .....(2)
From (1) and (2), we have
P(A) − P(A ∩ B) = 0 and P(B) − P(A ∩ B) = 0
⇒ P(A) = P(A ∩ B) and P(B) = P(A ∩ B)
⇒ P(A) = P(B)
APPEARS IN
RELATED QUESTIONS
A die is thrown. Describe the following events:
- A: a number less than 7
- B: a number greater than 7
- C: a multiple of 3
- D: a number less than 4
- E: an even number greater than 4
- F: a number not less than 3
Also find A ∪ B, A ∩ B, B ∪ C, E ∩ F, D ∩ E, A – C, D – E, E ∩ F', F'
Three coins are tossed once. Let A denote the event "three heads show", B denote the event "two heads and one tail show". C denote the event "three tails show" and D denote the event "a head shows on the first coin". Which events are
- mutually exclusive?
- simple?
- compound?
In a single throw of a die describe the event:
B = Getting a number greater than 7
In a single throw of a die describe the event:
C = Getting a multiple of 3
In a single throw of a die describe the event:
D = Getting a number less than 4
In a single throw of a die describe the event:
E = Getting an even number greater than 4
In a single throw of a die describe the event:
F = Getting a number not less than 3.
Also, find A ∪ B, A ∩ B, B ∩ C, E ∩ F, D ∩ F and \[ \bar { F } \] .
A and B are two events such that P (A) = 0.54, P (B) = 0.69 and P (A ∩ B) = 0.35. Find
P (A ∪ B).
A and B are two events such that P (A) = 0.54, P (B) = 0.69 and P (A ∩ B) = 0.35. Find
\[P (\bar{ A } \cap \bar{ B } )\]
A dice is thrown twice. What is the probability that at least one of the two throws come up with the number 3?
One number is chosen from numbers 1 to 100. Find the probability that it is divisible by 4 or 6?
If three dice are throw simultaneously, then the probability of getting a score of 5 is
An experiment consists of rolling a die until a 2 appears. How many elements of the sample space correspond to the event that the 2 appears not later than the k th roll of the die?
A team of medical students doing their internship have to assist during surgeries at a city hospital. The probabilities of surgeries rated as very complex, complex, routine, simple or very simple are respectively, 0.15, 0.20, 0.31, 0.26, .08. Find the probabilities that a particular surgery will be rated routine or complex
A team of medical students doing their internship have to assist during surgeries at a city hospital. The probabilities of surgeries rated as very complex, complex, routine, simple or very simple are respectively, 0.15, 0.20, 0.31, 0.26, .08. Find the probabilities that a particular surgery will be rated routine or simple
One urn contains two black balls (labelled B1 and B2) and one white ball. A second urn contains one black ball and two white balls (labelled W1 and W2). Suppose the following experiment is performed. One of the two urns is chosen at random. Next a ball is randomly chosen from the urn. Then a second ball is chosen at random from the same urn without replacing the first ball. Write the sample space showing all possible outcomes
One urn contains two black balls (labelled B1 and B2) and one white ball. A second urn contains one black ball and two white balls (labelled W1 and W2). Suppose the following experiment is performed. One of the two urns is chosen at random. Next a ball is randomly chosen from the urn. Then a second ball is chosen at random from the same urn without replacing the first ball. What is the probability that two black balls are chosen?
A card is drawn from a deck of 52 cards. Find the probability of getting a king or a heart or a red card.
A sample space consists of 9 elementary outcomes e1, e2, ..., e9 whose probabilities are
P(e1) = P(e2) = 0.08, P(e3) = P(e4) = P(e5) = 0.1
P(e6) = P(e7) = 0.2, P(e8) = P(e9) = 0.07
Suppose A = {e1, e5, e8}, B = {e2, e5, e8, e9}
Using the addition law of probability, calculate P(A ∪ B)
A sample space consists of 9 elementary outcomes e1, e2, ..., e9 whose probabilities are
P(e1) = P(e2) = 0.08, P(e3) = P(e4) = P(e5) = 0.1
P(e6) = P(e7) = 0.2, P(e8) = P(e9) = 0.07
Suppose A = {e1, e5, e8}, B = {e2, e5, e8, e9}
Calculate `P(barB)` from P (B), also calculate `P(barB)` directly from the elementary outcomes of `barB`
Determine the probability p, for the following events.
An odd number appears in a single toss of a fair die.
Determine the probability p, for the following events.
A king, 9 of hearts, or 3 of spades appears in drawing a single card from a well-shuffled ordinary deck of 52 cards.
Determine the probability p, for the following events.
The sum of 6 appears in a single toss of a pair of fair dice.
If P(A ∪ B) = P(A ∩ B) for any two events A and B, then ______.
The probability that at least one of the events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.2, then `P(barA) + P(barB)` is ______.
If M and N are any two events, the probability that at least one of them occurs is ______.
The probability of an occurrence of event A is 0.7 and that of the occurrence of event B is 0.3 and the probability of occurrence of both is 0.4
Let S = {1, 2, 3, 4, 5, 6} and E = {1, 3, 5}, then `barE` is ______.
