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Q12(ii):
Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (ii) at least one is a girl?
Solution :
Step 1: Define the Sample Space
Let $B$ denote a boy and $G$ denote a girl. For a family with two children, the sample space $S$ is the set of all possible ordered pairs $(c_1, c_2)$, where $c_1$ is the first child and $c_2$ is the second child.
$S = \{ (B, B), (B, G), (G, B), (G, G) \}$
The total number of equally likely outcomes is $n(S) = 4$.
Step 2: Identify Event Sets
Let $E$ be the event that both children are girls, and $F$ be the event that at least one child is a girl.
- Event $E$ (Both are girls): $E = \{ (G, G) \}$
- Event $F$ (At least one is a girl): $F = \{ (B, G), (G, B), (G, G) \}$
[By the definition of conditional probability, we require the intersection of these sets.]
Step 3: Intersection and Probabilities
Find the intersection $E \cap F$:
$E \cap F = \{ (G, G) \} \cap \{ (B, G), (G, B), (G, G) \} = \{ (G, G) \}$
The number of favorable outcomes are:
- $n(E \cap F) = 1$
- $n(F) = 3$
Since each outcome is equally likely with probability $P(\omega) = \frac{1}{4}$, the probabilities are:
$P(E \cap F) = \frac{n(E \cap F)}{n(S)} = \frac{1}{4}$
$P(F) = \frac{n(F)}{n(S)} = \frac{3}{4}$
Step 4: Applying the Conditional Probability Formula
The conditional probability $P(E|F)$ is defined as:
$P(E|F) = \frac{P(E \cap F)}{P(F)}$ [Per the definition of Conditional Probability]
Substituting the calculated values:
$P(E|F) = \frac{1/4}{3/4}$
$P(E|F) = \frac{1}{3}$
Visual Representation of Sample Space
Final Solution: The conditional probability that both children are girls given that at least one is a girl is $\frac{1}{3}$.
More Questions from Class 12 Mathematics Probability EXERCISE 13.1
- Q1: Given that E and F are events such that $P(E) = 0.6$, $P(F) = 0.3$ and $P(E \cap F) = 0.2$, find $P(E|F)$ and $P(F|E)$
- Q2: Compute $P(A|B)$, if $P(B) = 0.5$ and $P(A \cap B) = 0.32$
- Q3(i): If $P(A) = 0.8$, $P(B) = 0.5$ and $P(B|A) = 0.4$, find (i) $P(A \cap B)$
- Q3(ii): If $P(A) = 0.8$, $P(B) = 0.5$ and $P(B|A) = 0.4$, find (ii) $P(A|B)$
- Q3(iii): If $P(A) = 0.8$, $P(B) = 0.5$ and $P(B|A) = 0.4$, find (iii) $P(A \cup B)$
- Q4: Evaluate $P(A \cup B)$, if $2P(A) = P(B) = \frac{5}{13}$ and $P(A|B) = \frac{2}{5}$
- Q5(i): If $P(A) = \frac{6}{11}$ , $P(B) = \frac{5}{11}$ and $P(A \cup B) = \frac{7}{11}$, find (i) $P(A\cap B)$
- Q5(ii): If $P(A) = \frac{6}{11}$ , $P(B) = \frac{5}{11}$ and $P(A \cup B) = \frac{7}{11}$, find (ii) $P(A|B)$
- Q5(iii): If $P(A) = \frac{6}{11}$ , $P(B) = \frac{5}{11}$ and $P(A \cup B) = \frac{7}{11}$, find (iii) $P(B|A)$
- Q6(i): Determine $P(E|F)$. A coin is tossed three times, where (i) E : head on third toss, F : heads on first two tosses
- Q6(ii): Determine $P(E|F)$. A coin is tossed three times, where (ii) E : at least two heads, F : at most two heads
- Q6(iii): Determine $P(E|F)$. A coin is tossed three times, where (iii) E : at most two tails, F : at least one tail
- Q7(i): Determine $P(E|F)$. Two coins are tossed once, where (i) E : tail appears on one coin, F : one coin shows head
- Q7(ii): Determine $P(E|F)$. Two coins are tossed once, where (ii) E : no tail appears, F : no head appears
- Q8: Determine $P(E|F)$. A die is thrown three times, E : 4 appears on the third toss, F : 6 and 5 appears respectively on first two tosses
- Q9: Determine $P(E|F)$. Mother, father and son line up at random for a family picture E : son on one end, F : father in middle
- Q10(a): A black and a red dice are rolled. (a) Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5.
- Q10(b): A black and a red dice are rolled. (b) Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.
- Q11(i): A fair die is rolled. Consider events $E = \{1,3,5\}$, $F = \{2,3\}$ and $G = \{2,3,4,5\}$. Find (i) $P(E|F)$ and $P(F|E)$
- Q11(ii): A fair die is rolled. Consider events $E = \{1,3,5\}$, $F = \{2,3\}$ and $G = \{2,3,4,5\}$. Find (ii) $P(E|G)$ and $P(G|E)$
- Q11(iii): A fair die is rolled. Consider events $E = \{1,3,5\}$, $F = \{2,3\}$ and $G = \{2,3,4,5\}$. Find (iii) $P((E \cup F)|G)$ and $P((E \cap F)|G)$
- Q12(i): Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl,
- Q13: An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?
- Q14: Given that the two numbers appearing on throwing two dice are different. Find the probability of the event ‘the sum of numbers on the dice is 4’.
- Q15: Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event ‘the coin shows a tail’, given that ‘at least one die shows a 3’.
- Q16: If $P(A) = \frac{1}{2}$, $P(B) = 0$, then $P(A|B)$ is
- Q17: If A and B are events such that $P(A|B) = P(B|A)$, then
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