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To find the area of the shaded region, we need to subtract the area of the semi-circle from the area of the quadrant.

In a standard deck of 52 playing cards, there are 2 black kings: the king of spades and the king of clubs.

1. Total number of possible outcomes (total number of cards): There are 52 cards in a standard deck.

2. Total number of favorable outcomes (black kings): There are 2 black kings in the deck.
   

To find the probability that two different friends have different birthdays, we can consider the total number of possible combinations of birthdays and the number of combinations where the birthdays are different.

In a non-leap year, there are 365 days. Therefore, the probability that two friends have different birthdays is the complement of the probability that they have the same birthday.

1. Total number of possible combinations of birthdays: Each friend can have their birthday on any of the 365 days of the year. So, the total number of possible combinations of birthdays for two friends is 365×365365×365.

2. Number of combinations where the birthdays are different: If one friend has their birthday on a specific day, the other friend can have their birthday on any of the other 364 days of the year. So, the number of combinations where the birthdays are different is 365×364365×364.

Now, let's calculate the probability that two different friends have different birthdays:

Probability=Number of combinations where the birthdays are differentTotal number of possible combinations of birthdays

To find the probability of not drawing a white marble, we need to consider the total number of marbles in the box and the number of marbles that are not white.

Given:

• 3 blue marbles
• 2 white marbles
• 4 red marbles
  

In a standard deck of 52 playing cards, there are 2 black queens: the queen of spades and the queen of clubs.

1. Total number of possible outcomes (total number of cards): There are 52 cards in a standard deck.

2. Total number of favorable outcomes (black queens): There are 2 black queens in the deck.

Now, let's calculate the probability of drawing a black queen:

Probability=Number of favorable outcomesTotal number of possible outcomes

To find the probability of getting a black ball, we need to consider the total number of balls in the bag and the number of black balls.

To find the correct mean, we first need to adjust the sum of the observations by adding the difference between the correct value of 125 and the wrongly noted value of 25. Then we can divide this adjusted sum by the total number of observations.

To find the mean of the 32 numbers, we can use the concept of weighted averages.

Given:

1. The mean of 10 of the numbers is 15.
2. The mean of 20 of the numbers is 11.
3. The last two numbers are 10.

Let's denote:

The mean of the first n natural numbers can be calculated using the formula for the arithmetic mean:

To show that the mode of the combined series obtained by combining the two series S1 and S2 is different from that of S1 and S2 taken separately, we first need to find the mode of each series individually and then compare it with the mode of the combined series.