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Q9:
If A and B are two events such that $P(A) = \frac{1}{4}$, $P(B) = \frac{1}{2}$ and $P(A \cap B) = \frac{1}{8}$, find P (not A and not B).
Solution :
Step 1: Given Variables and Initial Setup
The problem provides the following probabilities for events $A$ and $B$ within a sample space $S$:
- $P(A) = \frac{1}{4}$
- $P(B) = \frac{1}{2}$
- $P(A \cap B) = \frac{1}{8}$
We are required to find the probability of the event "not $A$ and not $B$". In set-theoretic notation, this is represented as $P(A^c \cap B^c)$, where $A^c$ and $B^c$ are the complements of events $A$ and $B$, respectively.
Step 2: Applying De Morgan's Law
According to De Morgan’s Law in set theory, the intersection of the complements of two sets is equal to the complement of their union:
$A^c \cap B^c = (A \cup B)^c$
[Justification: By the Complement Law and De Morgan's Law of sets, $(A^c \cap B^c) = (A \cup B)^c$].
Therefore, we seek to calculate:
$P(A^c \cap B^c) = P((A \cup B)^c)$
Step 3: Utilizing the Complement Rule
The probability of the complement of an event is given by the formula:
$P(E^c) = 1 - P(E)$
[Per the Probability Axiom of Complementation], where $E = (A \cup B)$.
Thus, $P(A^c \cap B^c) = 1 - P(A \cup B)$.
Step 4: Calculating P(A ∪ B)
We apply the Addition Theorem of Probability to find the union of events $A$ and $B$:
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Substituting the given values into the equation:
$P(A \cup B) = \frac{1}{4} + \frac{1}{2} - \frac{1}{8}$
To perform the arithmetic, find the least common denominator (LCD), which is 8:
$P(A \cup B) = \frac{2}{8} + \frac{4}{8} - \frac{1}{8}$
$P(A \cup B) = \frac{2 + 4 - 1}{8} = \frac{5}{8}$
Step 5: Final Calculation
Returning to the expression derived in Step 3:
$P(A^c \cap B^c) = 1 - \frac{5}{8}$
$P(A^c \cap B^c) = \frac{8 - 5}{8} = \frac{3}{8}$
Final Solution: $P(\text{not } A \text{ and not } B) = \frac{3}{8}$
More Questions from Class 12 Mathematics Probability EXERCISE 13.2
- Q1: If $P(A) = \frac{3}{5}$ and $P(B) = \frac{1}{5}$, find $P(A \cap B)$ if A and B are independent events.
- Q2: 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.
- Q3: A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.
- Q4: 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.
- Q5: A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, ‘the number is even,’ and B be the event, ‘the number is red’. Are A and B independent?
- Q6: Let E and F be events with $P(E) = \frac{3}{5}$, $P(F) = \frac{3}{10}$ and $P(E \cap F) = \frac{1}{5}$. Are E and F independent?
- Q7(i): Given that the events A and B are such that $P(A) = \frac{1}{2}$, $P(A \cup B) = \frac{3}{5}$ and $P(B) = p$. Find $p$ if they are (i) mutually exclusive
- Q7(ii): Given that the events A and B are such that $P(A) = \frac{1}{2}$, $P(A \cup B) = \frac{3}{5}$ and $P(B) = p$. Find $p$ if they are (ii) independent.
- Q8(i): Let A and B be independent events with $P(A) = 0.3$ and $P(B) = 0.4$. Find (i) $P(A \cap B)$
- Q8(ii): Let A and B be independent events with $P(A) = 0.3$ and $P(B) = 0.4$. Find (ii) $P(A \cup B)$
- Q8(iii): Let A and B be independent events with $P(A) = 0.3$ and $P(B) = 0.4$. Find (iii) $P(A|B)$
- Q8(iv): Let A and B be independent events with $P(A) = 0.3$ and $P(B) = 0.4$. Find (iv) $P(B|A)$
- Q10: Events A and B are such that $P(A) = \frac{1}{2}$, $P(B) = \frac{7}{12}$ and P(not A or not B) = $\frac{1}{4}$. State whether A and B are independent ?
- Q11(i): Given two independent events A and B such that $P(A) = 0.3$, $P(B) = 0.6$. Find (i) P(A and B)
- Q11(ii): Given two independent events A and B such that $P(A) = 0.3$, $P(B) = 0.6$. Find (ii) P(A and not B)
- Q11(iii): Given two independent events A and B such that $P(A) = 0.3$, $P(B) = 0.6$. Find (iii) P(A or B)
- Q11(iv): Given two independent events A and B such that $P(A) = 0.3$, $P(B) = 0.6$. Find (iv) P(neither A nor B)
- Q12: A die is tossed thrice. Find the probability of getting an odd number at least once.
- Q13(i): Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (i) both balls are red.
- Q13(ii): Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (ii) first ball is black and second is red.
- Q13(iii): Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (iii) one of them is black and other is red.
- Q14(i): Probability of solving specific problem independently by A and B are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved
- Q14(ii): Probability of solving specific problem independently by A and B are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that (ii) exactly one of them solves the problem.
- Q15(i): One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent ? (i) E : ‘the card drawn is a spade’, F : ‘the card drawn is an ace’
- Q15(ii): One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent ? (ii) E : ‘the card drawn is black’, F : ‘the card drawn is a king’
- Q15(iii): One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent ? (iii) E : ‘the card drawn is a king or queen’, F : ‘the card drawn is a queen or jack’.
- Q16(a): In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random. (a) Find the probability that she reads neither Hindi nor English newspapers.
- Q16(b): In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random. (b) If she reads Hindi newspaper, find the probability that she reads English newspaper.
- Q16(c): In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random. (c) If she reads English newspaper, find the probability that she reads Hindi newspaper.
- Q17: The probability of obtaining an even prime number on each die, when a pair of dice is rolled is
- Q18: Two events A and B will be independent, if
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