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Q13:
Probability that A speaks truth is $\frac{4}{5}$. A coin is tossed. A reports that a head appears. The probability that actually there was head is
Solution :
Step 1: Defining Events and Given Probabilities
Let the events be defined as follows:
- Let $H$ be the event that a Head appears on the coin. $P(H) = \frac{1}{2}$
- Let $T$ be the event that a Tail appears on the coin. $P(T) = \frac{1}{2}$
- Let $E$ be the event that $A$ reports that a head appears.
The conditional probabilities regarding the truthfulness of $A$ are:
- Probability that $A$ speaks truth is $P(\text{Truth}) = \frac{4}{5}$
- Probability that $A$ lies is $P(\text{Lie}) = 1 - \frac{4}{5} = \frac{1}{5}$
Step 2: Calculating Conditional Likelihoods
Using the definition of truthfulness, we determine $P(E|H)$ and $P(E|T)$:
- $P(E|H)$ (A reports head when head actually appeared) = $P(\text{Truth}) = \frac{4}{5}$
- $P(E|T)$ (A reports head when tail actually appeared) = $P(\text{Lie}) = \frac{1}{5}$
Step 3: Applying Bayes' Theorem
We need to find $P(H|E)$, the probability that a Head appeared given that $A$ reported a Head. Per Bayes' Theorem [Theorem of Total Probability and Posterior Probability]:
$P(H|E) = \frac{P(H) \cdot P(E|H)}{P(H) \cdot P(E|H) + P(T) \cdot P(E|T)}$
Step 4: Substituting Values and Algebraic Simplification
Substitute the established probabilities into the theorem:
$P(H|E) = \frac{\left(\frac{1}{2}\right) \cdot \left(\frac{4}{5}\right)}{\left(\frac{1}{2}\right) \cdot \left(\frac{4}{5}\right) + \left(\frac{1}{2}\right) \cdot \left(\frac{1}{5}\right)}$
Factor out $\frac{1}{2}$ from the denominator:
$P(H|E) = \frac{\frac{1}{2} \cdot \frac{4}{5}}{\frac{1}{2} \cdot \left(\frac{4}{5} + \frac{1}{5}\right)}$
$P(H|E) = \frac{\frac{4}{5}}{\frac{5}{5}} = \frac{4}{5}$
Visual Representation of the Probability Space
Final Solution: The probability that actually there was a head is $\frac{4}{5}$.
More Questions from Class 12 Mathematics Probability EXERCISE 13.3
- Q1: An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?
- Q2: A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.
- Q3: Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostlier?
- Q4: In answering a question on a multiple choice test, a student either knows the answer or guesses. Let $\frac{3}{4}$ be the probability that he knows the answer and $\frac{1}{4}$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $\frac{1}{4}$. What is the probability that the student knows the answer given that he answered it correctly?
- Q5: A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive ?
- Q6: There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin ?
- Q7: An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?
- Q8: A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?
- Q9: Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.
- Q10: Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?
- Q11: A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?
- Q12: A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.
- Q14: If A and B are two events such that $A \subset B$ and $P(B) \neq 0$, then which of the following is correct?
CBSE Solutions for Class 12 Mathematics Probability
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