All questions of Probability for Mathematics Exam

Probability of having exactly one defective piece out of 5

- The probability of having a defective piece is 0.01, which means the probability of having a non-defective piece is 0.99.
- To find the probability of having exactly one defective piece out of 5 successive pieces, we need to use the binomial probability formula: P(X = k) = nCk * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and (1-p) is the probability of failure.
- In this case, n = 5, k = 1, p = 0.01, and (1-p) = 0.99.
- Plugging these values into the formula, we get: P(X = 1) = 5C1 * (0.01)^1 * (0.99)^(5-1) = 5 * 0.01 * (0.99)^4
- Therefore, the correct answer is option 'D': 5 * (0.99)^4 * (0.01)

Understanding the Problem:
We are given that a fair dice is rolled twice and we need to find the probability that an odd number will follow an even number.

Approach:
To solve this problem, we can use the concept of conditional probability. We know that the probability of an event A occurring given that event B has already occurred is given by P(A|B) = P(A ∩ B) / P(B), where P(A ∩ B) represents the probability of both events A and B occurring together, and P(B) represents the probability of event B occurring.

Calculating the Probability:
Let's consider event A as the occurrence of an odd number and event B as the occurrence of an even number in the first roll.

We know that the probability of rolling an odd number on a fair dice is 3/6 = 1/2 (since there are 3 odd numbers out of 6 possible outcomes). Similarly, the probability of rolling an even number on a fair dice is also 1/2 (since there are 3 even numbers out of 6 possible outcomes).

Now, let's calculate the probability of both events A and B occurring together (P(A ∩ B)). Since the outcomes of the two rolls are independent, the probability of both events occurring together is equal to the product of their individual probabilities. Therefore, P(A ∩ B) = P(A) * P(B) = (1/2) * (1/2) = 1/4.

Next, let's calculate the probability of event B occurring (P(B)), which is the probability of rolling an even number on the first roll. As mentioned earlier, this probability is 1/2.

Finally, we can use the formula for conditional probability to find the probability of event A occurring given that event B has already occurred. Using the formula P(A|B) = P(A ∩ B) / P(B), we have P(A|B) = (1/4) / (1/2) = 1/2.

Therefore, the probability that an odd number will follow an even number is 1/2, which corresponds to option 'D' in the given choices.

Understanding the Problem
When drawing two cards from a standard deck of 52 playing cards without replacement, we need to find the probability that both cards drawn are Kings.
Step 1: Total Number of Kings
- A standard deck has 4 Kings (one from each suit: hearts, diamonds, clubs, spades).
Step 2: Drawing the First Card
- The probability of drawing a King first:
- There are 4 Kings in a deck of 52 cards.
- Probability of drawing the first King = 4/52 = 1/13.
Step 3: Drawing the Second Card
- After drawing the first King, there are now:
- 3 Kings left in the deck.
- 51 cards remaining in total.
- The probability of drawing a second King:
- Probability of drawing the second King = 3/51.
Step 4: Calculating the Combined Probability
- To find the overall probability of both events (drawing two Kings):
- Multiply the probabilities of each event:
Probability = (Probability of first King) * (Probability of second King)
= (4/52) * (3/51)
= (1/13) * (1/17)
= 4/663.
Step 5: Simplifying the Probability
- To express 4/663 in a more useful manner:
- 4/663 simplifies and leads to the probability of drawing two Kings with no replacement being 1/221.
Final Answer
Thus, the correct answer is option 'D' (1/221), which illustrates the probability of drawing two Kings consecutively from a deck without replacement.

Explanation:

To find the probability of using a car, bus, and metro, we need to consider the different scenarios and calculate their probabilities.

Scenario 1: Using a private car
Given that the probability of using a private car is 0.45, the probability of using a car is 0.45.

Scenario 2: Using public transport
If the person decides to use public transport, there are two options available: bus and metro.

Scenario 2.1: Using a bus
Given that the probability of using a bus is 0.55, the probability of using a bus is 0.45 * 0.55 = 0.2475 (rounded to two decimal places).

Scenario 2.2: Using a metro
Since the person has already decided to use public transport and the only options left are bus and metro, the probability of using a metro is 1 - probability of using a bus = 1 - 0.2475 = 0.7525 (rounded to two decimal places).

Therefore, the probability of using a car, bus, and metro would be:
- Car: 0.45
- Bus: 0.2475 (rounded to two decimal places)
- Metro: 0.7525 (rounded to two decimal places)

Thus, the correct answer is option 'A': 0.45, 0.30, and 0.25.

Explanation:

To understand why option D is the correct answer, let's analyze the given information and use the method of least squares to find the regression lines.

1. Fitted line y = ax + b:
We are given that the fitted line when regressing y upon x is given by y = ax + b. This means that we are trying to find a linear relationship between the variables x and y, where a is the slope of the line and b is the y-intercept.

2. Fitted line x = cy + d:
We are also given that the fitted line when regressing x upon y is given by x = cy + d. In this case, we are trying to find a linear relationship between the variables y and x, where c is the slope of the line and d is the y-intercept.

Now, let's consider the relationship between these two lines:

- The slope of the first line, y = ax + b, is a.
- The slope of the second line, x = cy + d, is 1/c.

Since we are given that a is the slope of the first line, and the slope of the second line is 1/c, we can conclude that a must be equal to 1/c. Therefore, option D is correct.

Summary:
Based on the given information and the method of least squares, we can conclude that the slope of the fitted line when regressing y upon x (a) is equal to the reciprocal of the slope of the fitted line when regressing x upon y (1/c), which is represented by option D.

Coefficient of Variation (CV) is a measure of relative variability. It is used to compare the standard deviation of different datasets when the means are significantly different. It is calculated by dividing the standard deviation by the mean.

Given:
Standard deviation (σ) = 8.8 kmph
Mean (μ) = 33 kmph

To find the coefficient of variation, we use the formula:
CV = (σ / μ) * 100

Calculating the Coefficient of Variation:
CV = (8.8 / 33) * 100
CV = 0.2666 * 100
CV = 26.66

Since the coefficient of variation is expressed as a percentage, we can convert it to a decimal by dividing by 100:
CV = 26.66 / 100
CV = 0.2666

Therefore, the coefficient of variation in speed is 0.2666, which is option (c).

Explanation:
The coefficient of variation is a measure of the relative variability of a dataset. In this case, it is used to compare the standard deviation of the spot speeds of vehicles on the highway to the mean speed of the vehicles.

A lower coefficient of variation indicates that the dataset has less relative variability, while a higher coefficient of variation indicates greater relative variability.

In this case, the standard deviation is 8.8 kmph, which means that the spot speeds of vehicles on the highway vary by an average of 8.8 kmph from the mean speed of 33 kmph. Dividing the standard deviation by the mean gives us the coefficient of variation of 0.2666.

Option (c) is the correct answer.

The probability density function (PDF) given is:

f(t) = 1/t for -1 < t="" />< />

To determine the cumulative distribution function (CDF) F(t) for this continuous random variable, we need to integrate the PDF over the given range:

F(t) = ∫[from -∞ to t] f(u) du

For -1 < t="" />< 1,="" we="" />

F(t) = ∫[from -∞ to t] (1/u) du

To evaluate this integral, we need to split it into two parts:

F(t) = ∫[from -∞ to 0] (1/u) du + ∫[from 0 to t] (1/u) du

The first integral from -∞ to 0 is undefined because the PDF is not defined for negative values. Therefore, we can ignore this part.

For the second integral from 0 to t, we have:

F(t) = ∫[from 0 to t] (1/u) du

Now, to evaluate this integral, we can use the natural logarithm function:

F(t) = ln|u| [from 0 to t]

F(t) = ln|t| - ln|0|

Since ln|0| is undefined, we cannot evaluate it. However, as t approaches 0 from the positive side, ln|t| approaches -∞. Therefore, we can write:

F(t) = ln|t| for 0 ≤ t < />

So, the cumulative distribution function (CDF) for the given continuous random variable is:

F(t) = ln|t| for 0 ≤ t < 1="" />

Understanding the Problem
Manish has a maximum waiting time of 8 minutes at each of the two stops (B and C) and he can afford a total waiting time of 13 minutes. This means:
- Waiting at Stop B: X (0 to 8 minutes)
- Waiting at Stop C: Y (0 to 8 minutes)
The total waiting time is given by the equation:
X + Y > 13
We need to find the probability that Manish arrives late at D.
Setting Up the Scenario
- Both X and Y are uniformly distributed between 0 and 8.
- The total possible waiting time combinations (X, Y) form a square region in the XY-plane with vertices (0,0), (8,0), (0,8), and (8,8). The area of this square is 64 (8*8).
Finding the Area of Interest
1. Late Arrival Condition:
- The condition X + Y > 13 represents a line on the XY-plane.
2. Identifying the Intersection:
- The line intersects the boundaries of the square at points (5,8) and (8,5).
3. Calculating the Area Above the Line:
- The triangle formed by these points and the origin (0,0) has vertices (5,8), (8,5), and (8,8).
Calculating Areas
- Area of the triangle:
- Base = 8 - 5 = 3
- Height = 8 - 5 = 3
- Area = 0.5 * base * height = 0.5 * 3 * 3 = 4.5
- Total area above the line = Area of the square - Area of the triangle = 64 - 4.5 = 59.5
Probability Calculation
- Probability of arriving late (X + Y > 13):
P = Area above the line / Total area of the square = (64 - 4.5) / 64 = 59.5 / 64 = 8/13
Conclusion
Thus, the probability that Manish will arrive late at D is 8/13, making option 'A' the correct answer.