Probability

Let the probability of success be and the probability of failure be .
According to the given information, the probability of success is twice the probability of failure, so: Since the total probability is 1, we can express this as:
Substituting into the equation: Thus, .
Now, the probability of at most two failures in six trials is the sum of the probabilities of having 0, 1, or 2 failures.
This is a binomial probability problem, so we use the binomial distribution formula: For trials, the probability of at most two failures is:
We calculate each term:

1.

2.

3.

Thus, the total probability of at most two failures is:

Problem Analysis:
We need to find the probability of forming an isosceles triangle from 9 given points, where no three points are colinear.

1. Total Number of Triangles:
The number of triangles formed from 9 points is given by:

2. Number of Isosceles Triangles:
For a regular nonagon (9-sided polygon), the number of isosceles triangles is: This accounts for all possible isosceles configurations in the regular nonagon.

3. Probability Calculation:
The probability of randomly selecting an isosceles triangle is:

Final Answer:
The probability of forming an isosceles triangle is .

To determine the probability that a randomly chosen number from the set \{11, 12, 13, \ldots, 30\} is neither divisible by 3 nor divisible by 5, we can follow these steps:

Step 1: Determine the Total Number of Outcomes

The set contains numbers from 11 to 30, inclusive. To find the total number of outcomes, compute:

Thus, there are 20 numbers in the set.

Step 2: Determine Numbers Divisible by 3 or 5

Numbers Divisible by 3:

The numbers divisible by 3 are identified by solving within the range. Numbers satisfying this are 12, 15, 18, 21, 24, 27, 30.

Count: 7 numbers.

Numbers Divisible by 5:

The numbers divisible by 5 are identified by solving within the range. Numbers satisfying this are 15, 20, 25, 30.

Count: 4 numbers.

Step 3: Use Inclusion-Exclusion Principle

To find numbers that are either divisible by 3 or by 5, apply the inclusion-exclusion principle:

where:

= Numbers divisible by 3 = 7

= Numbers divisible by 5 = 4

= Numbers divisible by both, i.e., 15, 30

Count: 2 numbers.

Applying the principle:

This means there are 9 numbers divisible by either 3 or 5.

Step 4: Calculate the Probability

Numbers that are neither divisible by 3 nor by 5 are given by the total numbers minus those divisible by 3 or 5:

Total numbers = 20

Numbers neither divisible by 3 nor by 5 = 20 - 9 = 11

Thus, the probability is:

Therefore, the probability that a number is neither divisible by 3 nor divisible by 5 is .

Given the universal set , we analyze three subsets and their probabilities:

1. Set A Analysis:
Define .
Factorizing the quadratic:

Thus:

Count:
Probability:

2. Set B Analysis:
Define .
Primes in S:

Count:
Probability:

3. Set C Analysis:
Define .
Squares in S:

Count:
Probability:

4. Probability Comparison:

Final Answer:
The probability relationship is .

We have three cards with the following configurations:
1. First card: Both sides are red.
2. Second card: Both sides are black.
3. Third card: One side is red and the other side is black.
The three cards are mixed, and one card is picked randomly. We are given that the upper side of the chosen card is coloured red. We need to find the probability that the other side is coloured black. Let’s analyze the possible cases:
- Case 1: The chosen card is the first card (both sides red). In this case, the other side is also red.
- Case 2: The chosen card is the third card (one side red and one side black). In this case, the other side is black. Out of the three cards, two have a red side facing up (the first and third cards). The third card is the only one where the other side is black.
Thus, the probability that the other side is black, given that the upper side is red, is: Thus, the correct answer is .