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The correct choice is (c) POISSON distribution

Easiest EXPLANATION: Poisson distribution SHOWS the number of times an event is likely to occur within a specified time. The Poisson distribution probability FORMULA is P(x; μ) = (e^-μ) (μ^x) / x!

Right answer is (a) P(x; μ) = (e^-μ) (μ^x) / x!

The explanation: POISSON distribution shows the NUMBER of TIMES an event is likely to OCCUR within a specified time. The Poisson distribution probability formula is P(x; μ) = (e^-μ) (μ^x) / x!

Right choice is (b) 2

Easiest EXPLANATION: Bernoulli trial has only two possible outcomes and is mutually EXCLUSIVE. Those two outcomes are ‘success’ and ‘failure’. So, it is also CALLED as a ‘yes’ or ‘no’ QUESTION.

Right option is (B) \(\FRAC {4}{27}\)

The best explanation: Total number of balls = 9

The probability of the first ball to be red = \(\frac {3}{9}\)

The probability of the second ball to be green with REPLACEMENT = \(\frac {4}{9}\)

Total probability = \(\frac {3}{9} \times \frac {4}{9} = \frac {4}{27}\)

Correct choice is (C) 1/2

To EXPLAIN I would say: Possible outcomes when a DICE is thrown = {1,2,3,4,5,6}\(\frac {number \, of \, odd \, NUMBERS}{number \, of \, possible \, outcomes} = \frac {3}{6}=\frac {1}{2}\)

Correct answer is (b) \(\frac {1}{25}\)

Easiest explanation: PROBABILITY of YELLOW = \(\frac {1}{5}\)

Probability of choosing PINK pair of gloves twice = \(\frac {1}{5} \TIMES \frac {1}{5} = \frac {1}{25}\)

The correct ANSWER is (d) P(A∩B) = P(B) P(A/B)

To ELABORATE: The multiplication theorem of PROBABILITY is If A and B are two events of a random experiment with P(A) > 0 and P(B) > 0, then P(A∩B) = P(A) P(B/A) or P(A∩B) = P(B) P(A/B).

Correct OPTION is (c) 16/31

The explanation is: Let E1 = EVENT of choosing the bag 1, E2 = event of choosing the bag 2.

Let A be event of DRAWING a red BALL.

P(E1) = P(E2) = 1/2.

Also, P(A|E1) = P(drawing a red ball from Bag 1) = 3/8.

And P(A|E2) = P(drawing a red ball from Bag 2) = 4/10.

The probability of drawing a ball from bag 2, being GIVEN that it is red is P(E2|A).

By using Bayes’ theorem,

P(E2|A) = P(E2)P(A|E2)/( P(E1)P(A│E1)+P(E2)P(A|E2))

= (1/2 × 4/10) / ((1/2 × 3/8)) + (1/2 × 4/10)) = 16/31.

The correct option is (b) binomial DISTRIBUTION

To explain I would say: Formula for binomial distribution is P [X = x] = ^nCxp^x q^N-x

Where n is the number of TRIALS, x is the number of successes and (n-x) is failures.

The CORRECT answer is (a) \(\FRAC {1}{12}\)

Best explanation: The probability of coin LANDING on the tail = \(\frac {1}{2}\)

The probability of the dice showing 2 = \(\frac {1}{6}\)

TOTAL probability = \(\frac {1}{2} \TIMES \frac {1}{6} = \frac {1}{12}\)

Correct answer is (d) 0.5

To explain I would say: Total number of balls = 9

The PROBABILITY of the FIRST ball to be red = \(\frac {3}{9}\)

The probability of the second ball to be green WITHOUT REPLACEMENT = \(\frac {4}{8} = \frac {1}{2}\) = 0.5

Right answer is (c) Not defined

The BEST I can explain: We KNOW that P(A|B) = P(A∩B) / P(B). (By formula for conditional probability)

The value of P(B) = 0 in the GIVEN question. As the value of denominator BECOMES 0, the value of P(A|B) becomes un-defined.

The correct option is (c) 1/2

Best EXPLANATION: Possible OUTCOMES when a dice is thrown = {1, 2, 3, 4, 5, 6}

\(\FRAC {number \, of \, even \, numbers}{number \, of \, possible \, outcomes} = \frac {3}{6}=\frac {1}{2}\)

Correct answer is (c) \(\frac {1}{169}\)

To ELABORATE: The probability of drawing a FOUR = \(\frac {4}{52}\)

The probability of drawing a queen = \(\frac {4}{52}\)

Total probability = \(\frac {4}{52} \TIMES \frac {4}{52} = \frac {1}{169} \)

Correct option is (a) 1/8

Best explanation: We KNOW that ∑P(xi)=1

P(X) = P(X=0) + P(X=1) + P(X=2) + P(X=3)

1 = 0 + 1/2 + 2k + 2k

1/2 = 4k

k = 1/8

The correct answer is (a) If the OUTCOME of one event does not affect the outcome of another

Explanation: INDEPENDENT events refer to the outcome of one event does not affect the outcome of another. For example, if two PERSONS participating in two different RACES, WINNING one person doesn’t affect the winning of another person.

Right choice is (a) True

To ELABORATE: X = Number of days in a month.

The number of days in a month can be 29,30,31

X TAKES finite values so, therefore it is Discrete and RANGE X = {28,29,30,31}

Correct ANSWER is (c) \(\frac {1}{3}\)

For explanation: Total number of MARBLES = 7

Probability of the first MARBLE to be red = \(\frac {3}{7}\)

Probability of the SECOND marble to be red without replacement = \(\frac {2}{6} = \frac {1}{3}\)

The correct answer is (b) BAYES theorem

Explanation: Bayes theorem is the method in which the CALCULATED probabilities are revised with VALUES of new probabilities, whereas Updation theorem, REVISION theorem and Dependent theorem are not RELATED to the concept of probability.

Correct choice is (d) 1/3

For explanation: POSSIBLE outcomes when a dice is thrown = {1, 2, 3, 4, 5, 6}

Multiples of 3 = {3, 6}

Total PROBABILITY = \(\frac {number \, of \, multiples \, of \, 3}{number \, of \, possible \, outcomes} = \frac {2}{6}=\frac {1}{3}\)

Right ANSWER is (c) 1/6

Explanation: We KNOW that ∑P(XI)=1

 P(X) = P(X=0) + P(X=1) + P(X=2) + P(X=3)

1 = 0 + k + 2k + 3k

1 = 6k

K = 1/6

The correct CHOICE is (d) 0

The EXPLANATION: Yellow COLOR ball is not listed in the question. The bag contains only RED, white and green color BALLS. So, the probability to draw a yellow color ball out of the bag is zero.

The CORRECT option is (a) 2/3

The explanation is: We know that P(E|F) = P(E∩F) / P(F). (By FORMULA for conditional PROBABILITY)

VALUE of P(E∩F) is given to be 0.2 and value of P(F) is given to be 0.3.

P(E|F) = (0.2) / (0.3).

P(E|F) = 2 / 3.

Right option is (b) 1/4

For EXPLANATION: TOTAL number cards in a pack = 52

Total number of clubs in a pack of cards = 4

Total PROBABILITY = \(\frac {number \, of \, clubs}{number \, of \, cards \, in \, a \, pack} = \frac {13}{52}\)

Right choice is (b) 1/2

To explain I would SAY: We know that ∑P(XI) = 1

P(X) = P(X=0) + P(X=1) + P(X=2) + P(X=3)

1 = 1/4 + 1/4 + P(X=3)

P(X=3) = \(\frac {1}{2}\)

The CORRECT answer is (d) 7/12

The EXPLANATION: Let E1 = event of choosing the bag 1, E2 = event of choosing the bag 2.

Let A be event of DRAWING a black ball.

P(E1) = P(E2) = 1/2.

Also, P(A|E1) = P(drawing a black ball from Bag 1) = 6/10 = 3/5.

P(A|E2) = P(drawing a black ball from Bag 2) = 3/7.

By using BAYES’ theorem, the probability of drawing a black ball from bag 1 out of two bags is-:

P(E1|A) = P(E1)P(A|E1)/( P(E1)P(A│E1)+P(E2)P(A|E2))

= (1/2 × 3/5) / ((1/2 × 3/7)) + (1/2 × 3/5)) = 7/12.

The correct option is (B) P(A∩B) = P(A) P(B)

The EXPLANATION: Independent events refer to the OUTCOME of one EVENT does not affect the outcome of another. The FORMULA for independent events are P(A∩B) = P(A) P(B).

Right answer is (b) Dichotomous EXPERIMENT

Easy EXPLANATION: Bernoulli trials is ALSO CALLED a Dichotomous experiment and is repeated n TIMES. If in each trial the probability of success is constant, then such trials are called Bernoulli trails.

The correct choice is (c) \(\frac {1}{8}\)

Easiest explanation: Total number of balls = 9

The probability of the FIRST BALL to be white = \(\frac {2}{9}\)

The probability of DRAWING white ball again for the second TIME without replacement = \(\frac {1}{8}\)

Correct answer is (a) \(\FRAC {5}{11}\)

Explanation: Total number of balls = 5 + 7 = 12

The probability of the first ball to be blue = \(\frac {7}{12}\)

The probability of drawing red ball as the second ball out without replacement = \(\frac {5}{11}\)

The correct CHOICE is (b) 3/5

The explanation is: We know that P(F|E) = P(E∩F) / P(E). (By FORMULA for CONDITIONAL probability)

Value of P(E∩F) is given to be 0.3 and value of P(E) is given to be 0.5.

P(F|E) = (0.3) / (0.5).

P(F|E) = 3 / 5.

The correct CHOICE is (d) 5/8

Best explanation: F(1)=P(X ≤ 1)

P(X ≤ 1) = P(X=1) + P(X=0)

=1/2+ 1/8

= 5/8

Correct CHOICE is (a) P(A|B) = \(\frac{P(A∩B)}{P(B)}\)

For explanation I WOULD say: Conditional probability P(A | B) INDICATES the probability of event ‘A’ happening given that event B has happened.

Which in formula can be written as P(A|B) = \(\frac{P(A∩B)}{P(B)}\).

Whereas formula’s P(A|B) = \(\frac{P(A∩B)}{P(A)}\), P(A|B) = \(\frac{P(A)}{P(B)}\), P(A|B) = \(\frac{P(B)}{P(A)}\) doesn’t satisfies the specified conditions.

Correct answer is (d) \(\frac {1}{8}\)

To elaborate: Total number of balls = 9

The PROBABILITY of the FIRST ball to be white = \(\frac {2}{9}\)

The probability of DRAWING white ball again for the second time without replacement = \(\frac {1}{8}\)

Correct ANSWER is (b) 3/7

To explain: We know that P(A|B) = P(A∩B) / P(B). (By FORMULA for CONDITIONAL probability)

Which is equivalent to (3/13) / (7/13), HENCE the value of P(A|B) = 3/7.

Correct option is (B) True

To elaborate: Poisson distribution is used for INDEPENDENT events and it shows the number of times an EVENT is going to occur in SPECIFIED intervals of time. The FORMULA for Poisson distribution is P(x; μ) = (e^-μ) (μ^x) / x!

Right answer is (a) True

Easiest EXPLANATION: A RANDOM experiment that has only two mutually exclusive outcomes called ‘success’ and ‘FAILURE’. This experiment is called a Dichotomous experiment and is repeated N times. So, it is called as BERNOULLI trials.

The CORRECT CHOICE is (b) 1

The best I can explain: F(4)=P(X ≤ 4)

P(X ≤ 4) = P(X=4) + P(X=3) + P (X=2) + P(X=1) + P(X=0)

= 1/11 + 4/11 + 2/11 + 3/11 + 1/11

= 1

Correct ANSWER is (d) 3/8

Easiest explanation: Let E = EVENT that the man reports that six in the throwing of the die and let, S1 = event that six occurs and S2 = event that six does not occur.

P(S1) = Probability that six occurs = 1/6.

P(S2) = Probability that six does not occur = 5/6.

Also, P(E|S1) = P(Probability that man reports six occurs when six actually has occurred on the die) = 3/4.

P(E|S2) = P(Probability that man reports six occurs when six not actually occurred on the die) =

1 – 3/4 = 1/4.

By using Bayes’ theorem,

P(S2|E) = P(S1)P(E|S1)/(P(S1)P(E│S1)+P(S2)P(E|S2))

= (1/6 × 3/4) / ((1/6 × 3/4)) + (5/6 × 1/4)) = 3/8.

The CORRECT choice is (C) P [X = x] = ^NCX p^x q^N-x

For explanation I would say: Formula for binomial DISTRIBUTION is P [X = x] = ^nCxp^x q^n-x

Where n is the number of trials, x is the number of successes and (n-x) is failures.

Right choice is (c) Discrete probability distribution

Best explanation: Poisson distribution shows the number of times an EVENT is LIKELY to OCCUR within a specified time. It is USED only for independent EVENTS that occur at a constant rate within a given interval of time.