Probability

In this problem, we are asked to find the probability of getting an odd number when choosing a number randomly from the first 100 numbers (i.e., numbers from 1 to 100).

Step 1: Total Number of Possible Outcomes

The total number of outcomes in this case is the total number of numbers in the range from 1 to 100. This means there are 100 possible outcomes (since we are considering the first 100 numbers).

Step 2: Number of Odd Numbers

Odd numbers are those numbers that are not divisible by 2. The odd numbers between 1 and 100 are:

1, 3, 5, 7, ..., 97, 99

These odd numbers form a sequence where the first term is 1, the common difference is 2, and the last term is 99. The number of terms (odd numbers) in this sequence is 50.

Step 3: Probability Formula

The probability of an event is given by the formula:

Probability =

Step 4: Substituting Values

The number of odd numbers is 50, and the total number of possible outcomes is 100. So, the probability of getting an odd number is:

Probability =

Step 5: Simplification

The fraction can be simplified to . Therefore, the probability of getting an odd number is:

Probability =

Conclusion

So, the probability of randomly selecting an odd number from the first 100 numbers is , or 0.5.

For X = 2, the possible observations are (1, 2) and (2,1),
∴P_(X=2)= =
For X = 3, the possible observations are (1, 3), (3,1), (2,3) and (3, 2).
∴P_(X=3)= =
For X = 4, the possible observations are (1, 4), (4, 1), (2,4), (4,2), (3,4) and (4,3).
∴P_(X=4)= =
For X = 5, the possible observations are (1, 5), (5, 1), (2,5), (5,2), (3,5), (5,3) (5, 4) and (4,5).
∴P_(X=5)= =
For X = 6, the possible observations are (1, 6), (6, 1), (2,6), (6,2), (3,6), (6,3) (6, 4), (4,6), (5,6) and(6,5).P(X=6)=10/30=1/3.
Therefore, the required probability distribution is as follows.
Then, E(X)=∑X_iP(X_i)
E(X) = + + + +
E(X) = + + + +2
E(X) =
E(X) =
E(X) =

1. Understand the Problem:
We're given that X follows a binomial distribution with n = 5 trials and probability of success p, denoted as X ~ B(5, p). We're also given that P(X = 3) = 5P(X = 4). We need to find the variance of X.

2. Recall the Binomial Probability Formula:
The probability of getting exactly k successes in n trials is given by:
where .

3. Set Up the Given Equation:
We are given P(X = 3) = 5P(X = 4). Let's write out the probabilities using the binomial probability formula:



4. Substitute and Solve for p:
Now substitute these into the given equation:



Calculate the binomial coefficients:



Substitute these values back into the equation:



Now, we can simplify:

Divide both sides by 5: Divide both sides by (assuming ): Divide both sides by (assuming ):

5. Calculate the Variance:
The variance of a binomial distribution is given by:

Substitute and :



Therefore, the variance is .

1. Understand the Problem:
We're given that the mean plus the variance of a binomial distribution is 1.8. We're also given that n (the number of trials) is 5. We need to find p (the probability of success).

2. Recall Binomial Distribution Formulas:
Mean (μ): μ = np
Variance (σ²): σ² = np(1 - p)

3. Set Up the Equation:
We are given: Mean + Variance = 1.8
So, np + np(1 - p) = 1.8

4. Substitute the Given Value of n:
We know n = 5, so substitute that into the equation:


5. Simplify and Solve for p:
Multiply by 10 to eliminate the decimal: Divide by 2 to simplify:

6. Use the Quadratic Formula to Solve for p:
The quadratic formula is: In our equation, a = 25, b = -50, and c = 9.


7. Find the Two Possible Values of p:


8. Check Validity:
Since probability must be between 0 and 1, p = 0.2 is the only valid solution.

Therefore, the probability of success (p) is 0.2.

 In this case:

5. Simplify the Fraction:
Simplifying gives:

Therefore, the probability of drawing a black face card from a well-shuffled deck of 52 cards is .

Let's break down this problem step by step.
1. Define the Variables:

• Let 'F' represent a step forward.
• Let 'B' represent a step backward.
• Probability of a forward step
• Probability of a backward step
• Total number of steps = 11

2. Determine the Possible Scenarios:

• One step forward: 6 backward steps and 5 forward steps.
• One step backward: 6 forward steps and 5 backward steps.

3. Calculate the Number of Ways for Each Scenario:

• One step forward: The man needs 6 backward steps and 5 forward steps. The number of ways this can happen is given by the binomial coefficient:
• One step backward: The man needs 6 forward steps and 5 backward steps. The number of ways this can happen is also given by the binomial coefficient:

4. Calculate the Probability of Each Scenario:

• Probability of one step forward:
  which is approximately 0.170669.
• Probability of one step backward:
  which is approximately 0.115379.

5. Calculate the Total Probability:

• Total probability = 0.170669 + 0.115379
• Total Probability = 0.286048

6. Final Answer:
The probability that at the end of eleven steps, he is one step away from the starting point is approximately 0.2860.