Question:
Two factories \( F_1 \) and \( F_2 \) produce cricket bats that are labelled. Any randomly chosen bat produced by factory \( F_1 \) is defective with probability 0.5 and any randomly chosen bat produced by factory \( F_2 \) is defective with probability 0.1. One of the factories is chosen at random, and two bats are randomly purchased from the chosen factory. Let the labels on these purchased bats be \( B_1 \) and \( B_2 \). If \( B_1 \) is found to be defective, then the conditional probability that \( B_2 \) is also defective is equal to __________ (round off to 2 decimal places).
Two factories \( F_1 \) and \( F_2 \) produce cricket bats that are labelled. Any randomly chosen bat produced by factory \( F_1 \) is defective with probability 0.5 and any randomly chosen bat produced by factory \( F_2 \) is defective with probability 0.1. One of the factories is chosen at random, and two bats are randomly purchased from the chosen factory. Let the labels on these purchased bats be \( B_1 \) and \( B_2 \). If \( B_1 \) is found to be defective, then the conditional probability that \( B_2 \) is also defective is equal to __________ (round off to 2 decimal places).
Updated On: Jan 25, 2025
Show Solution
Verified By Collegedunia
Correct Answer: 0.42 - 0.44
Solution and Explanation
1. Define Events:
- Let \( F_1 \) and \( F_2 \) denote the factories chosen.
- Let \( D_1 \) and \( D_2 \) represent the events that \( B_1 \) and \( B_2 \) are defective, respectively.
2. Given Information:
- \( P(D_1 \mid F_1) = 0.5, \, P(D_2 \mid F_1) = 0.5 \).
- \( P(D_1 \mid F_2) = 0.1, \, P(D_2 \mid F_2) = 0.1 \).
- \( P(F_1) = P(F_2) = 0.5 \).
3. Using Total Probability:
- Compute \( P(D_1) \):
\[
P(D_1) = P(D_1 \mid F_1)P(F_1) + P(D_1 \mid F_2)P(F_2).
\]
Substituting values:
\[
P(D_1) = (0.5)(0.5) + (0.1)(0.5) = 0.25 + 0.05 = 0.3.
\]
4. Conditional Probability \( P(F_1 \mid D_1) \):
- By Bayes' theorem:
\[
P(F_1 \mid D_1) = \frac{P(D_1 \mid F_1)P(F_1)}{P(D_1)} = \frac{(0.5)(0.5)}{0.3} = \frac{0.25}{0.3} = \frac{5}{6}.
\]
5. Conditional Probability \( P(F_2 \mid D_1) \):
- Similarly:
\[
P(F_2 \mid D_1) = \frac{P(D_1 \mid F_2)P(F_2)}{P(D_1)} = \frac{(0.1)(0.5)}{0.3} = \frac{0.05}{0.3} = \frac{1}{6}.
\]
6. Conditional Probability \( P(D_2 \mid D_1) \):
- By the law of total probability:
\[
P(D_2 \mid D_1) = P(D_2 \mid F_1, D_1)P(F_1 \mid D_1) + P(D_2 \mid F_2, D_1)P(F_2 \mid D_1).
\]
- Since \( P(D_2 \mid F_1, D_1) = P(D_2 \mid F_1) = 0.5 \) and \( P(D_2 \mid F_2, D_1) = P(D_2 \mid F_2) = 0.1 \), we get:
\[
P(D_2 \mid D_1) = (0.5)\left(\frac{5}{6}\right) + (0.1)\left(\frac{1}{6}\right).
\]
- Simplify:
\[
P(D_2 \mid D_1) = \frac{5}{12} + \frac{1}{60} = 0.42 \text{ to } 0.44.
\]
Was this answer helpful?
0
0
Top IIT JAM MS Statistics Questions
- A bag has 5 blue balls and 15 red balls. Three balls are drawn at random from the bag simultaneously. Then the probability that none of the chosen balls is blue equals
- Let \( Y \) be a continuous random variable such that \( P(Y > 0) = 1 \) and \( \mathbb{E}(Y) = 1 \). For \( p \in (0, 1) \), let \( \xi_p \) denote the \( p \)th quantile of the probability distribution of the random variable \( Y \). Then which of the following statements is always correct?
- Let 𝑋 be a continuous random variable having the 𝑈(−2, 3) distribution. Then which of the following statements is correct?
- Let \( X \) be a random variable having the Poisson distribution with mean \( 1 \). Let \( g: \mathbb{N} \cup \{0\} \to \mathbb{R} \) be defined by
\[g(x) = \begin{cases} 1 & \text{if } x \in \{0, 2\} \\ 0 & \text{if } x \notin \{0, 2\} \end{cases}\]
Then \( E(g(X)) \) is equal to: - For \( n \in \mathbb{N} \), let \( Z_n \) be the smallest order statistic based on a random sample of size \( n \) from the \( U(0,1) \) distribution. Let \( nZ_n \xrightarrow{d} Z \) as \( n \to \infty \) for some random variable \( Z \). Then \( P(Z \leq \ln 3) \) is equal to:
View More Questions
Top IIT JAM MS Probability Questions
- A bag has 5 blue balls and 15 red balls. Three balls are drawn at random from the bag simultaneously. Then the probability that none of the chosen balls is blue equals
- Let \( Y \) be a continuous random variable such that \( P(Y > 0) = 1 \) and \( \mathbb{E}(Y) = 1 \). For \( p \in (0, 1) \), let \( \xi_p \) denote the \( p \)th quantile of the probability distribution of the random variable \( Y \). Then which of the following statements is always correct?
- For \( n \in \mathbb{N} \), let \( Z_n \) be the smallest order statistic based on a random sample of size \( n \) from the \( U(0,1) \) distribution. Let \( nZ_n \xrightarrow{d} Z \) as \( n \to \infty \) for some random variable \( Z \). Then \( P(Z \leq \ln 3) \) is equal to:
- Two fair coins \( S_1 \) and \( S_2 \) are tossed independently once. Let the events \( E, F, \) and \( G \) be defined as follows:
\( E \): Head appears on \( S_1 \)
\( F \): Head appears on \( S_2 \)
\( G \): The same outcome (head or tail) appears on both \( S_1 \) and \( S_2 \)
Then which of the following statements is NOT correct? - Let the random vector \( (X, Y) \) have the joint probability density function
\[f(x, y) = \begin{cases} \frac{1}{x}, & \text{if } 0 < y < x < 1 \\0, & \text{otherwise} \end{cases}.\]
Then \( \text{Cov}(X, Y) \) is equal to:
View More Questions
Top IIT JAM MS Questions
- Let \( a_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \) and \( b_n = \sum_{k=1}^{n} \frac{1}{k^2} \) for all \( n \in \mathbb{N} \). Then
- Let \( f(x,y) = x^2 + 3y^2 - \frac{2}{3} xy \) at \( (x,y) \in \mathbb{R}^2 \). Then
- Let \( A = \begin{pmatrix} a & 0 \\ c & d \end{pmatrix} \) be a real matrix, where \( ad = 1 \) and \( c \ne 0 \). If \( A^{-1} + (\text{adj} \, A)^{-1} = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \), then \( (\alpha, \beta, \gamma, \delta) \) is equal to
- A bag has 5 blue balls and 15 red balls. Three balls are drawn at random from the bag simultaneously. Then the probability that none of the chosen balls is blue equals
- Let \( Y \) be a continuous random variable such that \( P(Y > 0) = 1 \) and \( \mathbb{E}(Y) = 1 \). For \( p \in (0, 1) \), let \( \xi_p \) denote the \( p \)th quantile of the probability distribution of the random variable \( Y \). Then which of the following statements is always correct?
View More Questions