Advertisements
Advertisements
Question
If \[\frac{(1 - 3p)}{2}, \frac{(1 + 4p)}{3}, \frac{(1 + p)}{6}\] are the probabilities of three mutually exclusive and exhaustive events, then the set of all values of p is
Options
(0, 1)
(−1/4, 1/3)
(0, 1/3)
(0, ∞)
Solution
(−1/4, 1/3)
P(A) = (1 − 3p)/2
P(B) = (1 + 4p)/3
P(C) = (1 + p)/6
The events are mutually exclusive and exhaustive.
∴ P(A ∪ B ∪ C) = P(A) + P(B) + P(C) = 1
⇒\[0 \leq P\left( A \right) \leq 1\]
\[0 \leq \frac{1 + 4p}{3} \leq 1\]
\[0 \leq \frac{1 + p}{6} \leq 1\]
\[\Rightarrow - 1/3 \leq p \leq \frac{1}{3}\] , ...(i)
and \[- 1 \leq p \leq 5\] ...(iii)
The common solution of (i), (ii) and (iii) is\[- 1/4 \leq p \leq 1/3\]
APPEARS IN
RELATED QUESTIONS
A die is rolled. Let E be the event “die shows 4” and F be the event “die shows even number”. Are E and F mutually exclusive?
Three coins are tossed. Describe two events, which are not mutually exclusive.
Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice ≤ 5.
State true or false: (give reason for your answer).
A and B are mutually exclusive
Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice ≤ 5
State true or false: (give reason for your answer).
A = B'
Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice ≤ 5
State true or false: (give reason for your answer)
A and C are mutually exclusive
Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice ≤ 5
State true or false: (give reason for your answer)
A' , B' ,C are mutually exclusive and exhaustive.
Given P(A) = `3/5` and P(B) = `1/5`. Find P(A or B), if A and B are mutually exclusive events.
Events E and F are such that P(not E or not F) = 0.25, State whether E and F are mutually exclusive.
Three coins are tossed. Describe. two events A and B which are mutually exclusive.
Three coins are tossed. Describe. three events A, B and C which are mutually exclusive and exhaustive.
A die is thrown twice. Each time the number appearing on it is recorded. Describe the following events:
A = Both numbers are odd.
B = Both numbers are even.
C = sum of the numbers is less than 6
Also, find A ∪ B, A ∩ B, A ∪ C, A ∩ C
Which pairs of events are mutually exclusive?
Two dice are thrown. The events A, B, C, D, E and F are described as:
A = Getting an even number on the first die.
B = Getting an odd number on the first die.
C = Getting at most 5 as sum of the numbers on the two dice.
D = Getting the sum of the numbers on the dice greater than 5 but less than 10.
E = Getting at least 10 as the sum of the numbers on the dice.
F = Getting an odd number on one of the dice.
State true or false:
- A and B are mutually exclusive.
- A and B are mutually exclusive and exhaustive events.
- A and C are mutually exclusive events.
- C and D are mutually exclusive and exhaustive events.
- C, D and E are mutually exclusive and exhaustive events.
- A' and B' are mutually exclusive events.
- A, B, F are mutually exclusive and exhaustive events.
If A and B be mutually exclusive events associated with a random experiment such that P(A) = 0.4 and P(B) = 0.5, then find
P (A ∪ B)
If A and B be mutually exclusive events associated with a random experiment such that P(A) = 0.4 and P(B) = 0.5, then find
\[P (\bar{ A } \cap \bar{ B} )\]
If A and B be mutually exclusive events associated with a random experiment such that P(A) = 0.4 and P(B) = 0.5, then find
P ( \[\bar{ A} \] ∩ B)
Given two mutually exclusive events A and B such that P(A) = 1/2 and P(B) = 1/3, find P(A or B).
A box contains 10 white, 6 red and 10 black balls. A ball is drawn at random from the box. What is the probability that the ball drawn is either white or red?
In a race, the odds in favour of horses A, B, C, D are 1 : 3, 1 : 4, 1 : 5 and 1 : 6 respectively. Find probability that one of them wins the race.
A box contains 30 bolts and 40 nuts. Half of the bolts and half of the nuts are rusted. If two items are drawn at random, what is the probability that either both are rusted or both are bolts?
If A and B are mutually exclusive events such that P(A) = 0.35 and P(B) = 0.45, find P(A ∪ B)
If A and B are mutually exclusive events such that P(A) = 0.35 and P(B) = 0.45, find P(A ∩ B)
If A and B are mutually exclusive events such that P(A) = 0.35 and P(B) = 0.45, find P(A ∩ \[\bar{ B } \] )
The probabilities of three mutually exclusive events A, B and C are given by 2/3, 1/4 and 1/6 respectively. The statement
An experiment has four possible outcomes A, B, C and D, that are mutually exclusive. Explain why the following assignments of probabilities are not permissible:
P(A) = 0.12, P(B) = 0.63, P(C) = 0.45, P(D) = – 0.20
If A and B are any two events having P(A ∪ B) = `1/2` and P`(barA) = 2/3`, then the probability of `barA ∩ B` is ______.
A die is loaded in such a way that each odd number is twice as likely to occur as each even number. Find P(G), where G is the event that a number greater than 3 occurs on a single roll of the die.
If A and B are mutually exclusive events, P(A) = 0.35 and P(B) = 0.45, find P(B′)
If A and B are mutually exclusive events, P(A) = 0.35 and P(B) = 0.45, find P(A ∩ B)
If A and B are mutually exclusive events, P(A) = 0.35 and P(B) = 0.45, find P(A′ ∩ B′)
One of the four persons John, Rita, Aslam or Gurpreet will be promoted next month. Consequently the sample space consists of four elementary outcomes S = {John promoted, Rita promoted, Aslam promoted, Gurpreet promoted} You are told that the chances of John’s promotion is same as that of Gurpreet, Rita’s chances of promotion are twice as likely as Johns. Aslam’s chances are four times that of John. If A = {John promoted or Gurpreet promoted}, find P(A).
If A and B are mutually exclusive events, then ______.
| Column A | Column B |
| (a) If E1 and E2 are the two mutually exclusive events | (i) E1 ∩ E2 = E1 |
| (b) If E1 and E2 are the mutually exclusive and exhaustive events | (ii) (E1 – E2) ∪ (E1 ∩ E2) = E1 |
| (c) If E1 and E2 have common outcomes, then | (iii) E1 ∩ E2 = Φ, E1 ∪ E2 = S |
| (d) If E1 and E2 are two events such that E1 ⊂ E2 | (iv) E1 ∩ E2 = Φ |
Let E1, E2, E3 be three mutually exclusive events such that P(E1) = `(2 + 3p)/6`, P(E2) = `(2 - p)/8` and P(E3) = `(1 - p)/2`. If the maximum and minimum values of p are p1 and p2, then (p1 + p2) is equal to ______.
