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Right answer is (C) 3.7

To explain: EXPECTED Value: μ = E(X) = ∑x * P(x) = 4 × 0.32 + 5 × 0.47 = 3.63. Next find ∑x^2 * P(x): ∑x^2 * P(x) = 16 × 0.32 + 25 × 0.47 = 16.87. Therefore, VAR(X) = ∑x^2P(x) − μ^2 = 16.87 − 13.17 = 3.7.

The CORRECT choice is (C) 4.78

For EXPLANATION: We know that E(X) = ∑ x*P(x) = 2 × 0.45 + 4 × 0.97 = 4.78, where x={2,4}.

Right CHOICE is (a) \(\frac{13}{4}\)

The best I can explain: LET P(1) = P(2) = P(3) = P(4) = p; P(5) = P(6) = 2p. We know that the sum of all probabilities must be 1 ⇒ p + p + p + p + 2p + 2p = 1

 ⇒ 8p = 1 ⇒ p = \(\frac{1}{8}\)

 EXPECTED Value:

μ = E(X) = ∑x * P(x) =\(1 * \frac{1}{8} + 2 * \frac{1}{8} + 3 * \frac{1}{8} + 4 * \frac{1}{8} + 5 * \frac{2}{8} +6 * \frac{2}{8} = \frac{13}{4}\).

Correct choice is (c) 7

The best I can explain: SAMPLE space = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.Suppose: P(2) = \(\frac{1}{36}\), P(3) = \(\frac{2}{36}\), P(4) = \(\frac{3}{36}\), P(5) = \(\frac{4}{36}\), P(6) = \(\frac{5}{36}\), P(7) = \(\frac{6}{36}\), P(8) = \(\frac{5}{36}\), P(9) = \(\frac{4}{36}\), P(10) = \(\frac{3}{36}\), P(11) = \(\frac{2}{36}\) and P(12) = \(\frac{1}{36}\). Now, Expected VALUE:

μ = E(A) = ∑x * P(x) = \(2 * \frac{1}{36} + 3 * \frac{2}{36} + 4 * \frac{3}{36} + 5 * \frac{4}{36} + 6 * \frac{5}{36} \)

\(+ 7 * \frac{6}{36} + 8 * \frac{5}{36} + 9 * \frac{4}{36} + 10 * \frac{3}{36} + 11 * \frac{2}{36} + 12 * \frac{1}{36} = \frac{252}{36}\) = 7.

The correct CHOICE is (d) 2

For explanation: We know that E(X) = ∑{xi * P(xi)} = 3 * \(\frac{2}{13} + 7 * \frac{2}{13} − x * \frac{10}{13} = \frac{20}{13} − \frac{10x}{13}\). Suppose the expected value should be 0 Rs. for the game to be fair. So \(\frac{20}{13} − \frac{10x}{13}\) = 0 ⇒ x=2. So she should pay Rs.2 for it to be a fair game.

Right option is (C) 5.25

The BEST explanation: Multiply the OUTCOME by its probability, so the EXPECTED value becomes 0.75 * 7 POINTS = 5.25.

The correct choice is (B) ∫ f(x)DX = 1, -∞<=x<=∞

The explanation: A probability density function f(x) for the CONTINUOUS random variable X is DENOTED as ∫ f(x)dx = 1, -∞<=x<=∞. The area under the curve between any two ordinates x = a and x = b is a probability that X lies between a and b. So, ∫f(x)dx = P(a≤X≤b).

Right choice is (d) 1.5

The best explanation: Let H represents a head and T be a tail. X denotes the NUMBER of heads in three tosses of a coin. X can take the value 0, 1, 2, 3. P(X = 0) = \(\frac{1}{8}\), P(X = 1) = \(\frac{3}{8}\), P(X = 2) = \(\frac{3}{8}\), P(X = 3) = \(\frac{1}{8}\). The probability distribution of X is E(X) = Σixipi = 1 × \(\frac{3}{8} + 2 × \frac{3}{8} + 3 × \frac{1}{8}\) = 1.5. E(X2) = \(12 × \frac{3}{8} + 22 × \frac{3}{8} + 32 × \frac{1}{8}\) = 3. So, Variance of X = V(X) = E(X^2) – [E(X)]^2 = 3 – 1.5 = 1.5.

The correct answer is (C) 1.7 ≤ X < 2.9

For EXPLANATION: Possible outcomes should be 1.7 ≤ X < 2.9. That is the PROBABLE RANGE of X for the answer.

Correct choice is (a) 1

Explanation: Let X denote the number of red t-shirts in the OUTCOME. Here, x1 = 2, x2 = 1, x3 = 1, x4 = 1, x5 = 0. Probability of first t-shirt being red = \(\frac{5}{13}\).

Probability of second t-shirt being red = \(\frac{4}{12}\).

So: P(x1) = \(\frac{5}{13} × \frac{4}{12} = \frac{20}{146}\). Likewise, for the probability of red first FOLLOWED by black is \(\frac{8}{12}\) (as there are 8 red t-shirts still in the DRAWER and 12 t-shirts all together).

So, P(x2) = \(\frac{5}{13} *\frac{8}{12} = \frac{40}{146}\). Similarly for WHITE then red: P(x3) = \(\frac{8}{13} × \frac{4}{12} = \frac{32}{146}\).Finally, for 2 black balls: P(x4) = \(\frac{8}{13} × \frac{7}{12} = \frac{56}{146}\). So, \(\frac{20}{146} + \frac{40}{146} + \frac{32}{146} + \frac{40}{146} = 1\). Hence, all the t-shirts have been found.

Correct option is (a) data

To elaborate: We know that discrete probability function largely DEPENDS on the properties and TYPES of data such as Binomial distribution can lead to MODEL BINARY data such as flipping of coins.

Right option is (c) 42.1%

The best EXPLANATION: For x = 45000, Z = -2 and for x = 52000, z = 0.375. Now, area between z = -2 and z = 0.375 is EQUAL to 0.421 or 42.1% EARN between Rs. 45,000 and Rs. 52,000.

Correct choice is (C) 0.298

To elaborate: We have to find P(X < 25.4). Now, for x = 25.4, z becomes \(\frac{25.4 – 32}{3.6}\) = -1.83. HENCE, P(z < -1.83) = 0.298.

Right choice is (a) 0.341

To elaborate: The area is defined as the area under the standard normal curve.Now, for X = 32, z becomes \(\FRAC{32 – 43}{6.4}\) = -1.71. Hence, the required PROBABILITY is P(x < 32) = P(z < -1.71) = 0.341.

The CORRECT option is (d) 0.326

The EXPLANATION: We have to FIND P(6 < X < 11.6). Now, for x = 6, z becomes -1.062 and for z = 11.6, z = 0.687. So, P(6 < x < 11.6) = P(-1.062 < z < 0.687) = 0.326.

Correct choice is (b) 0.38

Explanation: Let x be the random variable that represents the speed of bicycle. x has μ = 90 and σ = 9.5. We have to FIND the probability that x is HIGHER than 95 or P(x > 95). For x = 95, Z = \(\frac{95 – 83}{9.4}\) = 1.27, P(x > 95) = P(z > 1.27) = [total area] – [area to the left of z = 1] = 1 – 0.620 = 0.38. The probability that a car selected at a random has a speed greater than 100 km/hr is EQUAL to 0.38.

The correct answer is (c) 0.0443

The EXPLANATION is: Suppose x be the random variable that represents the length of time. It has a MEAN of 66 and a standard deviation of 20. Find the PROBABILITY that x is between 70 and 90 or P(70 < x < 90). For x = 70, z = \(\frac{58 – 66}{20}\) = -4. For x = 75, z = \(\frac{75 – 66}{20}\) = 0.45. P(70 < x < 90) = P(-4 < z < 0.75) = [area to the left of z = 0.75] – [area to the left of z = -4] = 0.0443. The requiredprobability when the length of time between 58 and 75 hours is 0.0443.

The correct answer is (b) 0.2029

The best explanation: Let L be the random variable that represents the length of the COMPONENT. It has a mean of 7 CM and a standard deviation of 0.35 cm. To FIND P( 5.36 < X < 6.14). For x = 5.36, z = \(\frac{5.36 – 6}{0.35}\) = -1.82. For x = 6.14, z = \(\frac{6.14 – 6}{0.35}\) = 0.4 ⇒P(5.36 < x < 6.14) = P( -1.82 < z < 0.4) = 0.2029.

Correct answer is (a) \(\frac{1}{2}\)

For explanation: The sample space of the EXPERIMENT is S = {HH, HT, TH, TT}. Let X is the number of tails and It takes up the VALUES 0, 1 and 2. Now,P(no TAIL) = p(0) = \(\frac{1}{4}\), P(one tail) = p(1) = \(\frac{2}{4}\) and P(two tails) = p(2) = \(\frac{1}{4}\). So, X is the number of tail runs and it takes up the values 0 and 1. P(X = 0) = p(0) = \(\frac{2}{4} = \frac{1}{4}\).

The CORRECT answer is (a) \(\frac{5}{9}\)

To explain I would say: If the BALL is neither grey nor purple then it must be BLUE. There are 45 balls in TOTAL of which 25 are green and so the probability of picking a purple ball is \(\frac{25}{45} = \frac{5}{9}\).

The correct OPTION is (c) KL≤18.76

For explanation: The random SIDES of the triangle are K and L. These are the uniform random variables with uniform distributions on [0,8.3] and [0,4.7] RESPECTIVELY. They are independent and their JOINT distribution is uniform on the rectangle R = [0,8.3]∗[0,4.7]. The condition is KL/2≤9.38 ⇒ KL≤18.76. The probability that one needs is the ratio between the area under the hyperbola INSIDE R and the area of R.

Right option is (C) 28%

The explanation: we KNOW that the sum of the PROBABILITY that it RAINS and the probability that it does not rain must be 1. To determine the probability that it does not rain, CALCULATE 1 – 0.72 = 0.28.

The correct ANSWER is (d) 28%

To explain I would say: In this INSTANCE, a success is a BUG-free compilation, and a failure is the discovery of a bug. The programmer needs to have 0, 1 or 2 failures, so her PROBABILITY of finishing the program is: P(X=0) + P(X=1) + P(X=2) = (0.95)^0(0.1) + (0.95)^1(0.1) + (0.95)^2(0.1) = 0.28% = 28%.

Right ANSWER is (b) 0.95

The best explanation: IA success is a hit and a failure is a strike. The player requires either 0, 1, 2 or 3 failures in ORDER to GET a hit before STRIKING out, so the probability of a hit is:

P(X=0) + P(X=1) + P(X=2) + P(X=3) = (0.45)^0(0.55) + (0.45)^1(0.55) + (0.45)^2(0.55) + (0.45)^3(0.55) = 0.95.

Correct choice is (a) 54%

The best I can explain: The variance of a GEOMETRIC distribution with PARAMETER P is \(\FRAC{1-p}{p^2} = \frac{(1-0.72)}{0.722}\) = 0.54 or 54%. However, the variance of the geometric distribution and the variance of the shifted geometric distribution are identical.

The correct option is (b) 25%

Explanation: The set of outcomes are all of the points on the bin, which make up an area of where is the radius of the CIRCLE. The points which are closer to the center than to the edge are those that LIE within the circle of radius around the center. HENCE, the area of the success outcomes is π(r/2)^2 = πr^2/4. Thus, P(closer to center than edge)=(area of the desired outcome)/(area of the TOTAL outcome) = πr^2/4 /πr^2 = 1/4 = 0.25 = 25%.

The CORRECT CHOICE is (a) \(\frac{1}{273}\)

The best I can explain: To obtain a sum of 21 from three 7-sided dice is that 3 die will SHOW 7 face up. THEREFORE, the probability is simply \(\frac{1}{7} * \frac{1}{7} * \frac{1}{7} = \frac{1}{273}\).

Correct answer is (a) 12.5%

The best I can explain: Since there are infinitely many possible outcomes for the value of X we will take the EQUALLY LIKELY outcomes as RANDOM points along the number LINE from 5 to 9. R will be closer to 5 than it is to 6 if R<5.5. We can easily see it by drawing a probability line. Here, P(R is closer to 5 than to 6) = (length of segment where 5

Correct option is (d) \(\frac{1}{24}\)

The best I can explain: There is a \(\frac{1}{6}\) CHANCE of choosing the 4^thpath, and there is a \(\frac{1}{4}\) chance of choosing the 9^th path. The SELECTION of the path to the Kochi is independent of the selection of the path to the Trivendrum. Hence, by the rule of PRODUCT, there is a \(\frac{1}{6} * \frac{1}{4} = \frac{1}{24}\) chance of choosing the 4^th-9^th path.

Correct choice is (d) \(\FRAC{1}{52}\)

The EXPLANATION: The REQUIRED probability is : P(Jack or spade)=P(Jack)* P(spade) = \(\frac{4}{52} * \frac{13}{52} = \frac{1}{52}\).

The correct OPTION is (b) \(\frac{3}{197}\)

The best EXPLANATION: Using the rule of product, there are 196 possible combinations of rolls. Since having the red die = 3 and the BLUE die = 5 are one of the 196 combinations, the REQUIRED probability is \(\frac{1}{14}*\frac{1}{14} = \frac{1}{196}\).

Right option is (b) \(\frac{1}{126}\)

Easy explanation: There is a \(\frac{1}{3}\) probability that Mina WOULD randomly SELECT the orange scarf, a \(\frac{1}{6}\) probability to select the blue skirt, and a \(\frac{1}{7}\) probability to select the black top. These events are INDEPENDENT, that is, the selection of the scarf does not affect the selection of the tops and so on. HENCE, the probability that she SELECTS the clothings of her choice is \(\frac{1}{3} * \frac{1}{6} * \frac{1}{7}\) = 126.

Correct CHOICE is (d) 15

Explanation: Any positive DIVISOR of 4000 must be of the form 2^x5^y, where x and y are integers satisfying o<=x<=5 and 0<=y<=3. There are 5 POSSIBILITIES for x and 3 possibilities for y and HENCE there are 3*5 = 15(rule of product) positive divisors of 4000.

Right answer is (C) \(\frac{1}{64}\)

To explain: Each die roll is INDEPENDENT, that is, if the first die roll result is 8, it will not affect the probability of the second die roll resulting in 8. The probability of rolling one die is \(\frac{1}{8}\). Now, P (1st roll is 8 ∩ 2ND roll is 8). By using the rule of product: \(\frac{1}{8} * \frac{1}{8}\). Hence, the probability that both die ROLLS are 8 is \(\frac{1}{64}\).

Right answer is (d) 78.2

To ELABORATE: To count the number of integers = \(\FRAC{100}{2} + \frac{100}{4} + \frac{100}{5} – \frac{100}{8} – \frac{100}{20} + \frac{100}{100}\)

= 50 + 25 + 20 – 12.8 – 5 + 1 = 78.2.

Correct option is (a) 325

To ELABORATE: To choose any clock for the FIRST pick, there are 10+16=26 options. For the second choice, we have 25 clocks left to choose from and so on. Thus, by the rule of product, there are 26 * 25 * 24 * 23 = 650 POSSIBLE ways to choose exactly four clocks. However, we have counted every clock combination TWICE. Hence, the correct number of possible ways are 650/2 = 325.

The correct option is (d) 415

For explanation I WOULD say: The total number of journalists = 350.People who speak German = 275 -> people who do not speak German = 75, people who speak English = 250-> people who do not speak English = 100,people who speak Chinese = 200 -> people who do not speak Chinese = 150, people who speak Japanese = 260 -> people who do not speak Japanese = 90. The total number of people who do not KNOW at least one language will be maximum when the sets of people not knowing a particular language are mutually exclusive. Hence, the maximum number of people who do not know at least one language = 75 + 100 + 150 + 90 = 415.

The CORRECT choice is (d) 350

Easiest explanation: 50% PASSED in both the subjects, (75-50)% or 25% passed only in History and (65-50)% or 15% passed only in Mathematics, (50 + 25 + 15)% or 90% passed in both the subjects and 10% failed in both subjects. From the QUESTION, 10% of total candidates = 35. So, total candidates = 350.

Correct choice is (a) \(\FRAC{2}{3}\)

To explain: We can have, P(red)= \(\frac{1}{3}\), P(YELLOW) = \(\frac{1}{3}\),P(red or yellow)=P(red)+P(yellow) = \(\frac{1}{3} + \frac{1}{3} = \frac{2}{3}\).

The correct option is (b) \(\frac{43}{68}\)

The best I can explain: The “or” indicates FINDING the probability of a union of EVENTS. Let R be the event that a red marble is drawn and W be the event that a striped marble is drawn. R U W is the event that a marble that is either a red and a white striped is drawn. By the rule of sum of probability,

P(R U W) = P(R) + P(W) – p(R ⋂ W) = \(\frac{24}{68} + \frac{27}{68} – \frac{8}{68} = \frac{43}{68}\).

Hence, the probability of DRAWING a red or white striped marble is \(\frac{43}{68}\).

Right option is (d) \(\FRAC{7}{20}\)

The explanation is: Let X be the event that the number selected WOULD be divisible by 3 and Y be the event that the selected number would be divisible by 7. Then A u B denotes the event that the number would be divisible by 3 or 7. Now, X = {3, 9, 12, 15, 18} and Y = {7, 14} whereas S = {1, 2, 3, …,20}. Since A N B = Null set, so that the two events A and B are mutually EXCLUSIVE and as such we have,

P(A u B)=P(A) + P(B)⇒P(A u B)=\(\frac{5}{20} + \frac{2}{20}\)

Therefore,P(A u B)=\(\frac{7}{20}\).

The correct choice is (c) \(\frac{9}{20}\)

Easy EXPLANATION: There are 7 even NUMBERS on the 20-sided dice: 1, 3, 5, 7, 9, 13, 15. There are 6 prime numbers on the 20-sided dice: 2, 3, 5, 7, 11, 13. There are 4 numbers that are both odd and prime: 3, 5, 7, 13. By the rule of sum, the PROBABILITY that an odd or prime number is ROLLED is \((\frac{7}{20}) + (\frac{6}{20}) – (\frac{4}{20}) = \frac{9}{20}\).

Correct OPTION is (c) \(\frac{1}{2}\)

To EXPLAIN: Define the events A and B as follows: A=Neha chooses a black t-shirt. B= Neha chooses a blue skirt. Neha cannot CHOOSE both a black t-shirt and a blue t-shirt, so the addition theorem of PROBABILITY applies:

P(A U B) = P(A) + P(B) = \((\frac{6}{12}) + (\frac{2}{12}) = \frac{3}{6} = \frac{1}{2}\).