Probability

Step 1: Identifying Favorable Cases
A standard deck contains 52 cards, consisting of 4 suits: hearts, diamonds, clubs, and spades, each containing 13 cards.
- Number of diamond cards =
- Number of king cards =
- The king of diamonds is counted twice in both sets, so we subtract 1 to avoid double counting.
Step 2: Calculating Probability
The number of favorable cases:
Thus, the probability:

Step 1: Total Outcomes When Rolling Two Dice When two fair dice are rolled, each die has 6 faces, leading to a total number of possible outcomes:
Step 2: Favorable Outcomes Where Sum We list all possible pairs where the sum is greater than 7
- Sum = 8: (5 outcomes)
- Sum = 9: (4 outcomes) - Sum = 10: (3 outcomes)
- Sum = 11: (2 outcomes)
- Sum = 12: (1 outcome) Total favorable outcomes:
Step 3: Probability Calculation The probability of getting a sum greater than 7 is:

To solve the problem of finding the probability that the second ball is green, given that the first ball drawn was red, we need to process the events and their probabilities.

Step 1: Determine Total Initial Outcomes
The box initially contains 5 red balls and 4 green balls, so a total of 9 balls.

Step 2: Probability of First Ball Being Red
The probability of drawing a red ball first is given by:

Step 3: Probability of Second Ball Being Green Given First Is Red
After drawing one red ball, there are now 8 balls left in the box. Out of these, 4 are green. The probability that the second ball drawn is green, given the first was red, is:

Therefore, the probability that the second ball drawn is green, given that the first ball drawn was red, is .

To solve the problem of determining the probability that the difference between two randomly selected numbers from the first 6 natural numbers is less than 3, we follow these steps:

1. Identify the Sample Space:
   The first 6 natural numbers are: 1, 2, 3, 4, 5, 6. We are selecting 2 numbers without replacement, so the total number of ways to pick any 2 distinct numbers is given by . This is our sample space.
2. Calculate Favorable Outcomes:
   We need the difference between two selected numbers to be less than 3. Let's calculate the pairs:
   • Difference = 1: (2,1), (3,2), (4,3), (5,4), (6,5)
   • Difference = 2: (3,1), (4,2), (5,3), (6,4)
3. Calculate Probability:
   The probability is the ratio of the number of favorable outcomes to the total number of outcomes. Thus, the probability that the difference of the numbers is less than 3 is: .

Therefore, the probability that the difference between the two numbers is less than 3 is .

1. 3 Women and 2 Men:

   The number of ways to choose 3 women from 8 is given by the combination formula .

   The number of ways to choose 2 men from 10 is:

   Total for this scenario:

2. 4 Women and 1 Man:

   The number of ways to choose 4 women from 8:

   The number of ways to choose 1 man from 10:

   Total for this scenario:

3. 5 Women:

   The number of ways to choose 5 women from 8:

   Total for this scenario: 56

• Total committees with at least 3 women = 2520 + 700 + 56 = 3276