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Q12(ii):
Check whether the following probabilities $P(A)$ and $P(B)$ are consistently defined (ii) $P(A) = 0.5, P(B) = 0.4, P(A \cup B) = 0.8$
Solution :
Step 1: Identifying Mathematical Constraints for Consistent Probability
For any two events $A$ and $B$ in a sample space $S$, the probability axioms [Per Kolmogorov's Axioms] dictate the following conditions for consistency:
- $0 \leq P(A) \leq 1$ and $0 \leq P(B) \leq 1$
- $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
- Since $P(A \cap B) \geq 0$, it follows necessarily that $P(A \cup B) \leq P(A) + P(B)$ [By the Axiom of Additivity]
- Furthermore, $P(A \cup B)$ must be greater than or equal to the maximum of the individual probabilities, i.e., $P(A \cup B) \geq \max(P(A), P(B))$
Step 2: Analysis of Given Data
Given values:
- $P(A) = 0.5$
- $P(B) = 0.4$
- $P(A \cup B) = 0.8$
Step 3: Verification of Consistency Constraints
First, we determine the intersection $P(A \cap B)$ using the Addition Theorem of Probability:
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
$0.8 = 0.5 + 0.4 - P(A \cap B)$
$0.8 = 0.9 - P(A \cap B)$
$P(A \cap B) = 0.9 - 0.8 = 0.1$
Now, we verify the validity of $P(A \cap B)$:
- Since $0 \leq P(A \cap B) \leq 1$ and $0 \leq 0.1 \leq 1$, this condition is satisfied.
- Check constraint: $P(A \cap B) \leq \min(P(A), P(B))$
- $0.1 \leq \min(0.5, 0.4) \implies 0.1 \leq 0.4$ [This is True].
Step 4: Visualizing the Probability Space (Venn Diagram)
Step 5: Conclusion
Because all conditions of the probability axioms are satisfied—specifically that the intersection is non-negative and the union is less than or equal to the sum of individual probabilities—the defined probabilities are logically sound.
Final Solution: The probabilities $P(A) = 0.5, P(B) = 0.4$, and $P(A \cup B) = 0.8$ are consistently defined.
More Questions from Class 11 Mathematics Probability EXERCISE 14.2
- Q1: Which of the following can not be valid assignment of probabilities for outcomes of sample Space $S = \{\omega_1, \omega_2, \omega_3, \omega_4, \omega_5, \omega_6, \omega_7\}$
- Q2: A coin is tossed twice, what is the probability that atleast one tail occurs?
- Q3(i): A die is thrown, find the probability of following events: (i) A prime number will appear,
- Q3(ii): A die is thrown, find the probability of following events: (ii) A number greater than or equal to 3 will appear,
- Q3(iii): A die is thrown, find the probability of following events: (iii) A number less than or equal to one will appear,
- Q3(iv): A die is thrown, find the probability of following events: (iv) A number more than 6 will appear,
- Q3(v): A die is thrown, find the probability of following events: (v) A number less than 6 will appear.
- Q4(a): A card is selected from a pack of 52 cards. (a) How many points are there in the sample space?
- Q4(b): A card is selected from a pack of 52 cards. (b) Calculate the probability that the card is an ace of spades.
- Q4(c)(i): A card is selected from a pack of 52 cards. (c) Calculate the probability that the card is (i) an ace
- Q4(c)(ii): A card is selected from a pack of 52 cards. (c) Calculate the probability that the card is (ii) black card.
- Q5(i): A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed. find the probability that the sum of numbers that turn up is (i) 3
- Q5(ii): A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed. find the probability that the sum of numbers that turn up is (ii) 12
- Q6: There are four men and six women on the city council. If one council member is selected for a committee at random, how likely is it that it is a woman?
- Q7: A fair coin is tossed four times, and a person win Re 1 for each head and lose Rs 1.50 for each tail that turns up. From the sample space calculate how many different amounts of money you can have after four tosses and the probability of having each of these amounts.
- Q8(i): Three coins are tossed once. Find the probability of getting (i) 3 heads
- Q8(ii): Three coins are tossed once. Find the probability of getting (ii) 2 heads
- Q8(iii): Three coins are tossed once. Find the probability of getting (iii) atleast 2 heads
- Q8(iv): Three coins are tossed once. Find the probability of getting (iv) atmost 2 heads
- Q8(ix): Three coins are tossed once. Find the probability of getting (ix) atmost two tails
- Q8(v): Three coins are tossed once. Find the probability of getting (v) no head
- Q8(vi): Three coins are tossed once. Find the probability of getting (vi) 3 tails
- Q8(vii): Three coins are tossed once. Find the probability of getting (vii) exactly two tails
- Q8(viii): Three coins are tossed once. Find the probability of getting (viii) no tail
- Q9: If $\frac{2}{11}$ is the probability of an event, what is the probability of the event ‘not A’.
- Q10(i): A letter is chosen at random from the word ‘ASSASSINATION’. Find the probability that letter is (i) a vowel
- Q10(ii): A letter is chosen at random from the word ‘ASSASSINATION’. Find the probability that letter is (ii) a consonant
- Q11: In a lottery, a person choses six different natural numbers at random from 1 to 20, and if these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game? [Hint order of the numbers is not important.]
- Q12(i): Check whether the following probabilities $P(A)$ and $P(B)$ are consistently defined (i) $P(A) = 0.5, P(B) = 0.7, P(A \cap B) = 0.6$
- Q13(i): Fill in the blanks in following table: (i) $P(A) = \frac{1}{3}, P(B) = \frac{1}{5}, P(A \cap B) = \frac{1}{15}, P(A \cup B) = ...$
- Q13(ii): Fill in the blanks in following table: (ii) $P(A) = 0.35, P(B) = ..., P(A \cap B) = 0.25, P(A \cup B) = 0.6$
- Q13(iii): Fill in the blanks in following table: (iii) $P(A) = 0.5, P(B) = 0.35, P(A \cap B) = ..., P(A \cup B) = 0.7$
- Q14: Given $P(A) = \frac{3}{5}$ and $P(B) = \frac{1}{5}$. Find $P(A \text{ or } B)$, if A and B are mutually exclusive events.
- Q15(i): If E and F are events such that $P(E) = \frac{1}{4}$, $P(F) = \frac{1}{2}$ and $P(E \text{ and } F) = \frac{1}{8}$, find (i) $P(E \text{ or } F)$,
- Q15(ii): If E and F are events such that $P(E) = \frac{1}{4}$, $P(F) = \frac{1}{2}$ and $P(E \text{ and } F) = \frac{1}{8}$, find (ii) $P(\text{not } E \text{ and not } F)$.
- Q16: Events E and F are such that $P(\text{not } E \text{ or not } F) = 0.25$, State whether E and F are mutually exclusive.
- Q17(i): A and B are events such that $P(A) = 0.42, P(B) = 0.48$ and $P(A \text{ and } B) = 0.16$. Determine (i) $P(\text{not } A)$,
- Q17(ii): A and B are events such that $P(A) = 0.42, P(B) = 0.48$ and $P(A \text{ and } B) = 0.16$. Determine (ii) $P(\text{not } B)$ and
- Q17(iii): A and B are events such that $P(A) = 0.42, P(B) = 0.48$ and $P(A \text{ and } B) = 0.16$. Determine (iii) $P(A \text{ or } B)$
- Q18: In Class XI of a school 40% of the students study Mathematics and 30% study Biology. 10% of the class study both Mathematics and Biology. If a student is selected at random from the class, find the probability that he will be studying Mathematics or Biology.
- Q19: In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first examination is 0.8 and the probability of passing the second examination is 0.7. The probability of passing atleast one of them is 0.95. What is the probability of passing both?
- Q20: The probability that a student will pass the final examination in both English and Hindi is 0.5 and the probability of passing neither is 0.1. If the probability of passing the English examination is 0.75, what is the probability of passing the Hindi examination?
- Q21(i): In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at random, find the probability that (i) The student opted for NCC or NSS.
- Q21(ii): In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at random, find the probability that (ii) The student has opted neither NCC nor NSS.
- Q21(iii): In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at random, find the probability that (iii) The student has opted NSS but not NCC.
CBSE Solutions for Class 11 Mathematics Probability
Chapters in CBSE - Class 11 Mathematics
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