Probability Theory: Combinatorial Tasks with Solutions

Task:
When two dice are rolled simultaneously, calculate the probability that the sum of the dice is exactly 7.
Step-by-step Solution:
1. Determine possible pairs: The pairs \(1,6\), \(2,5\), \(3,4\), \(4,3\), \(5,2\), \(6,1\) yield a sum of 7.
2. Number of favorable outcomes: There are 6 favorable combinations.
3. Total number of possible outcomes: Each die has 6 sides → 6 × 6 = 36 possible combinations.
4. Calculate probability: 6 favorable cases / 36 total cases = \( \frac{1}{6} \).
Answer:
The probability of rolling a sum of 7 with two dice is 1/6.
⚙ augensumme

Task:
In classic lottery, 6 balls are drawn from 49. The question is: What is the probability of matching all 6 numbers?
Step-by-step Solution:
1. Calculate combinations: Drawing 6 numbers from 49 is done without order and without replacement.
2. Formula for combinations: \(C(n, k) = \frac{n!}{k! \cdot (n-k)!}\)
3. Insert the numbers: \( C(49, 6) = \frac{49!}{6! \cdot 43!} = 13983816 \)
4. Calculate probability: There is exactly 1 correct combination out of 13,983,816 possibilities.
5. Result: Probability = \(\frac{1}{13\,983\,816}\) ≈ 0.0000000715
Answer:
The probability of getting a "six" in lottery is 1 in 13,983,816.
⚙ lotto

Task:
Two cards are drawn randomly from a poker deck of 52 cards. What is the probability that both drawn cards are aces?
Step-by-step Solution:
1. Number of aces in the deck: There are 4 aces in a deck of cards.
2. Combinations for 2 aces: \(C(4, 2) = 6\) ways to choose 2 aces.
3. Combinations for 2 arbitrary cards: \(C(52, 2) = 1326\) ways to choose 2 cards from 52.
4. Calculate probability: \(\frac{6}{1326} = \frac{1}{221} \)
5. Decimal result: ≈ 0.0045
Answer:
The probability of randomly drawing 2 cards from a poker deck and getting two aces is about 1 in 221.
⚙ karten

Question:
An urn contains 3 red and 5 blue balls. Two balls are drawn in succession with replacement. What is the probability of drawing a red ball twice?
Step-by-step Solution:
1. Probability for a red ball per draw: \(\frac{3}{8}\)
2. Because drawing is with replacement, the probability remains the same.
3. For two reds: \(P = \frac{3}{8} \times \frac{3}{8} = \frac{9}{64}\)
4. Decimal value: ≈ 0.1406
Answer:
The probability of drawing a red ball twice in succession with replacement is \( \frac{9}{64} \) or about 14.1%.
⚙ kugeln

Question:
Suppose 5 students are to be seated on 5 chairs. How many possibilities are there if the order matters?
Step-by-step Solution:
1. Permutations count all arrangements when order matters: \( n! \)
2. Plug in for 5 students: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
3. This means: There are 120 different ways to seat the 5 students.
Note: If not all seats are filled (e.g., 3 out of 5), you would calculate \( P(5,~3) = 5 \times 4 \times 3 = 60\) possibilities.
Answer:
With all seats occupied, there are 120 different seating arrangements.
⚙ permutation