NCERT Solutions for Class 12 Maths Chapter 13 - Probability Exercise 13.3

Exercise 13.3

1. 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?

Ans: Given the urn contains 5 red and 5 black balls.

Let a red ball be drawn in the first attempt.

Then, probability of drawing a red ball \[=\dfrac{5}{10}=\dfrac{1}{2}\]

Now, we add two more red balls. Therefore the urns contain 7 red and 5 black balls.

Probability of drawing the second red ball \[=\dfrac{7}{12}\]

Let a black ball be drawn in the first attempt.

Then, probability of drawing a black ball \[=\dfrac{5}{10}=\dfrac{1}{2}\]

Now, we add two more red balls. Therefore the urns contain 5 red and 7 black balls.

Probability of drawing the second red ball \[=\dfrac{5}{12}\]

Therefore, total probability of drawing second ball as red is

$\dfrac{1}{2}\times \dfrac{7}{12}+\dfrac{1}{2}\times \dfrac{5}{12}=\dfrac{1}{2}\left( \dfrac{7}{12}+\dfrac{5}{12} \right)$

$=\dfrac{1}{2}\times 1$

$ =\dfrac{1}{2}$

2. 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.

Ans: Let A be the event of getting a red ball.

Let \[{{E}_{1}}\] and \[{{E}_{2}}\] be the events of selecting the first bag and second bag respectively.

\[P({{E}_{1}})=P({{E}_{2}})=\dfrac{1}{2}\]

\[\Rightarrow P(A|{{E}_{1}})=\]P(drawing a red ball from first bag) \[=\dfrac{4}{8}=\dfrac{1}{2}\]

\[\Rightarrow P(A|{{E}_{2}})=\]P(drawing a red ball from second bag) \[=\dfrac{2}{8}=\dfrac{1}{4}\]

The probability of drawing a ball from the first bag, 

given that it is red, is given by \[P({{E}_{1}}|A)\]

By using bayes theorem, we obtain

$P({{E}_{1}}|A)=\dfrac{P({{E}_{1}})P(A|{{E}_{1}})}{P({{E}_{1}})P(A|{{E}_{1}})+P({{E}_{2}})P(A|{{E}_{2}})}$

$=\dfrac{\dfrac{1}{2}\times \dfrac{1}{2}}{\dfrac{1}{2}\times \dfrac{1}{2}+\dfrac{1}{2}\times \dfrac{1}{4}}$

$=\dfrac{\dfrac{1}{4}}{\dfrac{1}{4}+\dfrac{1}{8}}$

$=\dfrac{2}{3}$

3. 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 hostler?

Ans: Let \[{{E}_{1}}\] and \[{{E}_{2}}\] be the events where the student is a hostler and a day scholar respectively.

Let A be the event that the chosen students get a grade A.

\[P({{E}_{1}})=60%=0.6\]

\[P({{E}_{2}})=40%=0.4\]

\[\Rightarrow P(A|{{E}_{1}})=\]P(student getting an A grade is a hostler) \[=30%=0.3\]

\[\Rightarrow P(A|{{E}_{2}})=\]P(student getting an A grade is a day scholar) \[=20%=0.2\]

The probability that a randomly chosen student is a hostler, given that he has an A grade, is given by \[P({{E}_{1}}|A)\]

By using Bayes theorem, we get,

$P({{E}_{1}}|A)=\dfrac{P({{E}_{1}})P(A|{{E}_{1}})}{P({{E}_{1}})P(A|{{E}_{1}})+P({{E}_{2}})P(A|{{E}_{2}})}$

$=\dfrac{0.6\times 0.3}{0.6\times 0.3+0.4\times 0.2}$

$=\dfrac{0.18}{0.26}$

$=\dfrac{9}{13}$

4. In answering a question on a multiple choice test, a student either knows the answer or guesses.  Let \[\dfrac{3}{4}\] be the probability that he knows the answer and \[\dfrac{1}{4}\] be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability \[\dfrac{1}{4}\] . What is the probability that the student knows the answer given that he answered it correctly?

Ans: Let \[{{E}_{1}}\] and \[{{E}_{2}}\] be the events in which the student knows the answer and the student guesses the answer respectively.

Let A be the event that the answer is correct.

\[P({{E}_{1}})=\dfrac{3}{4}\]

\[P({{E}_{2}})=\dfrac{1}{4}\]

\[\Rightarrow P(A|{{E}_{1}})=\]P(student answer correctly, given he knows the answer) \[=1\]

\[\Rightarrow P(A|{{E}_{2}})=\]P(student answer correctly, given that he guessed) \[=\dfrac{1}{4}\]

The probability that the student knows the answer, given that the answer is correct, is given by \[P({{E}_{1}}|A)\]

By using Bayes theorem, we get,

$P({{E}_{1}}|A)=\dfrac{P({{E}_{1}})P(A|{{E}_{1}})}{P({{E}_{1}})P(A|{{E}_{1}})+P({{E}_{2}})P(A|{{E}_{2}})}$

$=\dfrac{\dfrac{3}{4}\times 1}{\dfrac{3}{4}\times 1+\dfrac{1}{4}\times \dfrac{1}{4}}$

$=\dfrac{\dfrac{3}{4}}{\dfrac{3}{4}+\dfrac{1}{16}}$

$=\dfrac{3}{4}\div \dfrac{13}{16}$

$=\dfrac{12}{13}$

5. 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 (that is, if a healthy person is tested, then, with probability0.005, the test will imply he has the disease). If 0.1percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

Ans: Let \[{{E}_{1}}\] and \[{{E}_{2}}\] be the respective events that a person has a disease and a person has no disease.

\[P({{E}_{1}})=0.1%=0.001\]

Since \[{{E}_{1}}\] and \[{{E}_{2}}\] are events complementary to each other

\[P({{E}_{2}})=1-P({{E}_{1}})=1-0.001=0.999\]

Let A be the event that the answer is correct.

\[\Rightarrow P(A|{{E}_{1}})=\]P(result is positive given the person has disease) \[=99%=0.99\]

\[\Rightarrow P(A|{{E}_{2}})=\]P(result is positive given that the person has no disease) \[=0.5%=0.005\]

The probability that a person has a disease, given that his test result is positive, is given by \[P({{E}_{1}}|A)\]

By using Bayes theorem, we get,

$P({{E}_{1}}|A)=\dfrac{P({{E}_{1}})P(A|{{E}_{1}})}{P({{E}_{1}})P(A|{{E}_{1}})+P({{E}_{2}})P(A|{{E}_{2}})}$

$=\dfrac{0.001\times 0.99}{0.001\times 0.99+0.999\times 0.005}$

$=\dfrac{0.00099}{0.00099+0.004995}$

$=\dfrac{0.00099}{0.005985}$

$=\dfrac{22}{133}$

6. There are three coins. One is 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?

Ans: Let \[{{E}_{1}}\], \[{{E}_{2}}\] and \[{{E}_{3}}\]be the events of choosing a two headed coin, a biased coin, and an unbiased coin respectively.

Let A be the event that the coin shows heads.

\[\therefore P({{E}_{1}})=P({{E}_{2}})=P({{E}_{3}})=\dfrac{1}{3}\]

A two-headed coin will always show heads.

\[\Rightarrow P(A|{{E}_{1}})=\]P(coin showing heads, given that it is two-headed coin) \[=1\]

Probability of getting heads, given that the coin is biased \[=75%\]

\[\Rightarrow P(A|{{E}_{2}})=\]P(coin showing heads, given that the coin is biased) \[=\dfrac{75}{100}=\dfrac{3}{4}\]

The third coin is unbiased hence the probability of getting heads is \[=\dfrac{1}{2}\]

\[\Rightarrow P(A|{{E}_{3}})=\]P(coin showing heads, given that the coin is unbiased) \[=\dfrac{1}{2}\]

The probability that the coin is two-headed, given that it shows heads, is given by \[P({{E}_{1}}|A)\]

By using Bayes theorem, we get,

$P({{E}_{1}}|A)=\dfrac{P({{E}_{1}})P(A|{{E}_{1}})}{P({{E}_{1}})P(A|{{E}_{1}})+P({{E}_{2}})P(A|{{E}_{2}})+P({{E}_{3}})P(A|{{E}_{3}})}$

$=\dfrac{\dfrac{1}{3}\times 1}{\dfrac{1}{3}\times 1+\dfrac{1}{3}\times \dfrac{3}{4}+\dfrac{1}{3}\times \dfrac{1}{2}}$

$=\dfrac{\dfrac{1}{3}}{\dfrac{1}{3}\left( 1+\dfrac{3}{4}+\dfrac{1}{2} \right)}$

$=\dfrac{1}{9}\div 4$

$=\dfrac{4}{9}$

7. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of 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?

Ans: Let \[{{E}_{1}}\], \[{{E}_{2}}\] and \[{{E}_{3}}\]be the events that the driver is a scooter driver, a car driver, and a truck driver respectively.

Total number of drivers = 2000 scooter drivers + 4000 car drivers + 6000 truck drivers = 12000 drivers

\[\therefore P({{E}_{1}})=\dfrac{2000}{12000}=\dfrac{1}{6}\]

\[P({{E}_{2}})=\dfrac{4000}{12000}=\dfrac{1}{3}\]

\[PP({{E}_{3}})=\dfrac{6000}{12000}=\dfrac{1}{2}\]

Let A be the event that the person meets with an accident.

\[\Rightarrow P(A|{{E}_{1}})=\]P(scooter driver met with an accident) \[=0.01=\dfrac{1}{100}\]

\[\Rightarrow P(A|{{E}_{2}})=\]P(car driver met with an accident) \[=0.03=\dfrac{3}{100}\]

\[\Rightarrow P(A|{{E}_{3}})=\]P(truck driver met with an accident) \[=0.15=\dfrac{15}{100}\]

The probability that the driver is a scooter driver, given that he met with an accident, is given by \[P({{E}_{1}}|A)\]

By using Bayes theorem, we get,

$P({{E}_{1}}|A)=\dfrac{P({{E}_{1}})P(A|{{E}_{1}})}{P({{E}_{1}})P(A|{{E}_{1}})+P({{E}_{2}})P(A|{{E}_{2}})+P({{E}_{3}})P(A|{{E}_{3}})}$

$=\dfrac{\dfrac{1}{6}\times \dfrac{1}{100}}{\dfrac{1}{6}\times \dfrac{1}{100}+\dfrac{1}{3}\times \dfrac{3}{100}+\dfrac{1}{2}\times \dfrac{15}{100}}$

$=\dfrac{\dfrac{1}{6}\times \dfrac{1}{100}}{\dfrac{1}{100}\left( \dfrac{1}{6}+1+\dfrac{15}{2} \right)}$

$=\dfrac{1}{6}\div \dfrac{104}{12}$

$=\dfrac{1}{52}$

8. 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. Future, 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 random from this and is found to be defective. What is the probability that was produced by machine B?

Ans: Let \[{{E}_{1}}\] and \[{{E}_{2}}\] be the respective events of items produced by machines A and B.

\[\therefore P({{E}_{1}})=60%=\dfrac{3}{5}\]

\[P({{E}_{2}})=40%=\dfrac{2}{5}\]

Let A be the event that the produced items were found to be defective.

\[\Rightarrow P(A|{{E}_{1}})=\]P(product is defective, given that machine A produced) \[=2%=\dfrac{2}{100}\]

\[\Rightarrow P(A|{{E}_{2}})=\]P(product is defective, given that machine B produced) \[=1%=\dfrac{1}{100}\]

The probability that the randomly selected items was from B, given that it is defective, is given by \[P({{E}_{2}}|A)\]

By using Bayes theorem, we get,

$P({{E}_{2}}|A)=\dfrac{P({{E}_{2}})P(A|{{E}_{2}})}{P({{E}_{1}})P(A|{{E}_{1}})+P({{E}_{2}})P(A|{{E}_{2}})}$

$=\dfrac{\dfrac{2}{5}\times \dfrac{1}{100}}{\dfrac{3}{5}\times \dfrac{2}{100}+\dfrac{2}{5}\times \dfrac{1}{100}}$

$=\dfrac{\dfrac{2}{500}}{\dfrac{6}{500}+\dfrac{2}{500}}$

$=\dfrac{2}{8}$

$=\dfrac{1}{4}$

9. 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.

Ans: Let \[{{E}_{1}}\] and \[{{E}_{2}}\] be the respective events in which the first group and the second group win the competition.

\[\therefore P({{E}_{1}})=0.6\]

\[P({{E}_{2}})=0.4\]

Let A be the event of introducing a new product.

\[\Rightarrow P(A|{{E}_{1}})=\]P(introducing a new product, given that first group wins) \[=0.7\]

\[\Rightarrow P(A|{{E}_{2}})=\]P(introducing a new product, given that second group wins) \[=0.3\]

The probability that the new product is introduced by the second group, is given by \[P({{E}_{2}}|A)\]

By using Bayes theorem, we get,

$P({{E}_{2}}|A)=\dfrac{P({{E}_{2}})P(A|{{E}_{2}})}{P({{E}_{1}})P(A|{{E}_{1}})+P({{E}_{2}})P(A|{{E}_{2}})}$

$=\dfrac{0.4\times 0.3}{0.6\times 0.7+0.4\times 0.3}$

$=\dfrac{0.12}{0.42+0.12}$

$=\dfrac{0.12}{0.54}$

$=\dfrac{12}{54}$

$=\dfrac{2}{9}$

10. 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?

Ans: Let \[{{E}_{1}}\] be the event that the outcome on the die is 5 or 6 and \[{{E}_{2}}\] be the event that the outcome on the die is 1,2,3, or 4.

\[\therefore P({{E}_{1}})=\dfrac{2}{6}=\dfrac{1}{3}\]

\[P({{E}_{2}})=\dfrac{4}{6}=\dfrac{2}{3}\]

Let A be the event of getting exactly one head.

\[P({{E}_{1}})\] is same as tossing the coin 3 times, similarly \[P({{E}_{2}})\] is same as tossing the coin exactly once

\[\Rightarrow P(A|{{E}_{1}})=\]P(getting exactly one head, given the coin is tossed 3 times) \[=\dfrac{3}{8}\](TTH,THT,HTT)

\[\Rightarrow P(A|{{E}_{2}})=\]P(getting exactly one head, given the coin is tossed only once) \[=\dfrac{1}{2}\]

The probability that the girl threw 1,2,3, or 4 with die, if she obtained exactly one head, is given by \[P({{E}_{2}}|A)\]

By using Bayes theorem, we get,

$P({{E}_{2}}|A)=\dfrac{P({{E}_{2}})P(A|{{E}_{2}})}{P({{E}_{1}})P(A|{{E}_{1}})+P({{E}_{2}})P(A|{{E}_{2}})}$

$=\dfrac{\dfrac{2}{3}\times \dfrac{1}{2}}{\dfrac{1}{3}\times \dfrac{3}{8}+\dfrac{2}{3}\times \dfrac{1}{2}}$ 

$=\dfrac{\dfrac{1}{3}}{\dfrac{1}{3}\left( \dfrac{3}{8}+1 \right)}$

$=\dfrac{1}{11}\div 8$

$=\dfrac{8}{11}$

11. A manufacture has three machine operators A, B and C. The first operator A produces 1% defective items, whereas 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 was produced by A?

Ans: Let \[{{E}_{1}}\], \[{{E}_{2}}\] and \[{{E}_{3}}\]be the events that the time consumed by machine operator A, B and C on the job.

\[\therefore P({{E}_{1}})=50%=\dfrac{50}{100}=\dfrac{1}{2}\]

\[P({{E}_{2}})=30%=\dfrac{30}{100}=\dfrac{3}{10}\]

\[P({{E}_{3}})=20%=\dfrac{20}{100}=\dfrac{1}{5}\]

Let A be the event of producing defective items.

\[\Rightarrow P(A|{{E}_{1}})=\]P(item is defective, given that it is produced by A) \[=1%=\dfrac{1}{100}\]

\[\Rightarrow P(A|{{E}_{2}})=\]P(item is defective, given that it is produced by B) \[=5%=\dfrac{5}{100}\]

\[\Rightarrow P(A|{{E}_{3}})=\]P(item is defective, given that it is produced by C) \[=7%=\dfrac{7}{100}\]

The probability that the defective item was produced by A is given by, is given by \[P({{E}_{1}}|A)\]

By using Bayes theorem, we get,

$P({{E}_{1}}|A)=\dfrac{P({{E}_{1}})P(A|{{E}_{1}})}{P({{E}_{1}})P(A|{{E}_{1}})+P({{E}_{2}})P(A|{{E}_{2}})+P({{E}_{3}})P(A|{{E}_{3}})}$

$=\dfrac{\dfrac{1}{2}\times \dfrac{1}{100}}{\dfrac{1}{2}\times \dfrac{1}{100}+\dfrac{3}{10}\times \dfrac{5}{100}+\dfrac{1}{5}\times \dfrac{7}{100}}$

$=\dfrac{\dfrac{1}{100}\times \dfrac{1}{2}}{\dfrac{1}{100}\left( \dfrac{1}{2}+\dfrac{3}{2}+\dfrac{7}{2} \right)}$

$=\dfrac{1}{2}\div \dfrac{17}{5}$

$=\dfrac{5}{34}$

12. 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.

Ans: Let \[{{E}_{1}}\] and \[{{E}_{2}}\] be the respective events of choosing a diamond card and a card which is not diamond.

Let A be the event that denote the lost card

\[\therefore P({{E}_{1}})=\dfrac{13}{52}=\dfrac{1}{4}\]

\[P({{E}_{2}})=\dfrac{39}{52}=\dfrac{3}{4}\]

When the card lost is a diamond card, then there are 12 diamond cards out of 51 cards.

From 12 diamond cards 2 cards can be drawn in \[{}^{12}{{C}_{2}}\] ways.

Therefore the probability of getting two diamond cards, when one diamond card is lost, is given by \[P(A|{{E}_{1}})\].

\[P(A|{{E}_{1}})=\dfrac{{}^{12}{{C}_{2}}}{{}^{51}{{C}_{2}}}=\dfrac{12!}{2!\times 10!}\times \dfrac{2!\times 49!}{5!}=\dfrac{22}{425}\]

When the card lost is not a diamond card, then there are 13 diamond cards out of 51 cards.

From 13 diamond cards 2 cards can be drawn in \[{}^{13}{{C}_{2}}\] ways.

Therefore the probability of getting two diamond cards, when the card lost is not a diamond, is given by \[P(A|{{E}_{2}})\].

\[P(A|{{E}_{2}})=\dfrac{{}^{13}{{C}_{2}}}{{}^{51}{{C}_{2}}}=\dfrac{13!}{2!\times 11!}\times \dfrac{2!\times 49!}{5!}=\dfrac{26}{425}\]

The probability that the lost card is diamond is given by \[P({{E}_{1}}|A)\].

$P({{E}_{1}}|A)=\dfrac{P({{E}_{1}})P(A|{{E}_{1}})}{P({{E}_{1}})P(A|{{E}_{1}})+P({{E}_{2}})P(A|{{E}_{2}})}$

$=\dfrac{\dfrac{1}{4}\times \dfrac{22}{425}}{\dfrac{1}{4}\times \dfrac{22}{425}+\dfrac{3}{4}\times \dfrac{26}{425}}$

$=\dfrac{11}{2}\div 25$

$=\dfrac{11}{50}$

13. Probability that A speaks truth is \[\dfrac{4}{5}\] . A coin is tossed. A reports that a head appears. The probability that actually there was head is

A) \[\dfrac{4}{5}\]

B) \[\dfrac{1}{2}\]

C) \[\dfrac{1}{5}\]

D) \[\dfrac{2}{5}\]

Ans: Let \[{{E}_{1}}\] and \[{{E}_{2}}\] be the events such that

\[{{E}_{1}}\] : A speaks truth 

\[{{E}_{2}}\] : A speaks false 

Let X be the event that a head appears.

\[P({{E}_{1}})=\dfrac{4}{5}\] 

\[\therefore P({{E}_{2}})=1-P({{E}_{1}})=1-\dfrac{4}{5}=\dfrac{1}{5}\] 

If a coin is tossed, then it may result in either head(H) or tail(T).

The probability of getting a head is \[\dfrac{1}{2}\] whether A speaks truth or not.

\[\therefore P(X|{{E}_{1}})=P(X|{{E}_{2}})=\dfrac{1}{2}\]

The probability that there actually a head is given by \[P({{E}_{1}}|X)\].

$P({{E}_{1}}|X)=\dfrac{P({{E}_{1}})P(X|{{E}_{1}})}{P({{E}_{1}})P(X|{{E}_{1}})+P({{E}_{2}})P(X|{{E}_{2}})}$

$=\dfrac{\dfrac{4}{5}\times \dfrac{1}{2}}{\dfrac{4}{5}\times \dfrac{1}{2}+\dfrac{1}{5}\times \dfrac{1}{2}}$ 

$=\dfrac{\dfrac{4}{5}\times \dfrac{1}{2}}{\dfrac{1}{2}\left( \dfrac{4}{5}+\dfrac{1}{5} \right)}$

$=\dfrac{4}{5}\div 1$

$=\dfrac{4}{5}$

14. If A and B are two events such that \[A\subset B\] and \[P(B)\ne 0\], then which of the following is correct?

A) \[P(A|B)=\dfrac{P(B)}{P(A)}\]

B) \[P(A|B)<P(A)\]

C) \[P(A|B)\ge P(A)\]

D) None of these

Ans: If \[A\subset B\] then \[A\cap B=A\] ,

 \[\Rightarrow A\cap B=P(A)\]

Also, \[P(A)<P(B)\]

Consider \[P(A|B)=\dfrac{P(A\cap B)}{P(B)}=\dfrac{P(A)}{P(B)}\ne \dfrac{P(B)}{P(A)}\]     - (Eq 1)

Consider \[P(A|B)=\dfrac{P(A\cap B)}{P(B)}=\dfrac{P(A)}{P(B)}\]              - (Eq 2)

It is known that, \[P(B)\le 1.\]

$\Rightarrow \dfrac{1}{P(B)}\ge$

$\Rightarrow \dfrac{P(A)}{P(B)}\ge P(A)$

From (2), we obtain

\[\Rightarrow P(A|B)\ge P(A)\]          - (Eq 3)

\[\therefore P(A|B)\] is not less than \[P(A)\].

Thus, from (3), it can be concluded that the relation given in alternate C is correct.

Conclusion

NCERT Solutions for Maths Exercise 13.3 Class 12 Chapter 13 - Probability by Vedantu help you understand Bayes' theorem clearly. Focus on how to use this theorem to solve tricky probability questions. The step-by-step solutions make learning easier. Practicing these problems will boost your understanding and help you do better in exams.

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