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Q10:
Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.
Solution :
Step 1: Definition of Events and Initial Probabilities
Let the events be defined as follows:
- $E_1$: The ball transferred from Bag I to Bag II is red.
- $E_2$: The ball transferred from Bag I to Bag II is black.
- $A$: The ball drawn from Bag II is red.
Bag I contains 3 red balls and 4 black balls (Total = 7 balls).
Bag II contains 4 red balls and 5 black balls (Total = 9 balls).
Probabilities of transferring balls from Bag I:
- $P(E_1) = \frac{3}{7}$
- $P(E_2) = \frac{4}{7}$
Step 2: Conditional Probabilities of Drawing a Red Ball
If $E_1$ occurs, Bag II receives a red ball. The composition of Bag II becomes (4+1) red and 5 black (Total = 10 balls).
$P(A|E_1) = \frac{5}{10} = \frac{1}{2}$
If $E_2$ occurs, Bag II receives a black ball. The composition of Bag II becomes 4 red and (5+1) black (Total = 10 balls).
$P(A|E_2) = \frac{4}{10} = \frac{2}{5}$
Step 3: Application of Bayes' Theorem
We are required to find the probability that the transferred ball was black, given that the drawn ball is red, i.e., $P(E_2|A)$.
Per Bayes' Theorem [Theorems of Total Probability and Bayes' Rule]:
$P(E_2|A) = \frac{P(E_2) \cdot P(A|E_2)}{P(E_1) \cdot P(A|E_1) + P(E_2) \cdot P(A|E_2)}$
Step 4: Algebraic Computation
Substitute the established values into the formula:
$P(E_2|A) = \frac{\frac{4}{7} \cdot \frac{4}{10}}{\left(\frac{3}{7} \cdot \frac{5}{10}\right) + \left(\frac{4}{7} \cdot \frac{4}{10}\right)}$
$P(E_2|A) = \frac{\frac{16}{70}}{\frac{15}{70} + \frac{16}{70}}$
$P(E_2|A) = \frac{\frac{16}{70}}{\frac{31}{70}}$
$P(E_2|A) = \frac{16}{31}$
Final Solution: The probability that the transferred ball is black is $\frac{16}{31}$.
More Questions from Class 12 Mathematics Probability Miscellaneous Exercise on Chapter 13
- Q1(i): $A$ and $B$ are two events such that $P(A) \neq 0$. Find $P(B|A)$, if (i) $A$ is a subset of $B$
- Q1(ii): $A$ and $B$ are two events such that $P(A) \neq 0$. Find $P(B|A)$, if (ii) $A \cap B = \phi$
- Q2(i): A couple has two children, (i) Find the probability that both children are males, if it is known that at least one of the children is male.
- Q2(ii): A couple has two children, (ii) Find the probability that both children are females, if it is known that the elder child is a female.
- Q3: Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.
- Q4: Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed?
- Q5: If a leap year is selected at random, what is the chance that it will contain 53 tuesdays?
- Q6: Suppose we have four boxes A,B,C and D containing coloured marbles as given below: One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A?, box B?, box C?
- Q7: Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
- Q8: If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability $\frac{1}{2}$).
- Q9(i): An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known: $P(A \text{ fails}) = 0.2$, $P(B \text{ fails alone}) = 0.15$, $P(A \text{ and } B \text{ fail}) = 0.15$. Evaluate the following probabilities (i) $P(A \text{ fails|B has failed})$
- Q9(ii): An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known: $P(A \text{ fails}) = 0.2$, $P(B \text{ fails alone}) = 0.15$, $P(A \text{ and } B \text{ fail}) = 0.15$. Evaluate the following probabilities (ii) $P(A \text{ fails alone})$
- Q11: If A and B are two events such that $P(A) \neq 0$ and $P(B | A) = 1$, then
- Q12: If $P(A|B) > P(A)$, then which of the following is correct :
- Q13: If A and B are any two events such that $P(A) + P(B) – P(A \text{ and } B) = P(A)$, then
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