Explanation:

The total number of outcomes in the sample space when flipping n fair coins is given by 2^n. This is because for each coin, there are two possible outcomes - either a head or a tail. Since there are n coins, the total number of outcomes is 2 * 2 * ... * 2 (n times) = 2^n.

Probability of getting exactly r-number of heads:

To find the probability of getting exactly r-number of heads when n coins are tossed, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes.

Number of favorable outcomes:

The number of favorable outcomes is equal to the number of ways we can choose r coins out of n coins to be heads. This is represented by the combination formula nCr, which calculates the number of combinations of n items taken r at a time.

The combination formula nCr is given by n! / (r! * (n-r)!) where "!" represents the factorial of a number. The factorial of a number is the product of all positive integers less than or equal to that number.

In this case, the number of favorable outcomes is nCr.

Total number of possible outcomes:

As mentioned earlier, the total number of outcomes in the sample space is 2^n, which represents all possible combinations of heads and tails for n coins.

Probability:

The probability of getting exactly r-number of heads is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.

Therefore, the probability can be calculated as follows:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
= nCr / 2^n

This formula calculates the probability of obtaining exactly r heads when flipping n fair coins. It takes into account the number of ways we can choose r coins to be heads and divides it by the total number of possible outcomes.

By using the combination formula and the total number of outcomes formula, we can determine the probability of various outcomes when flipping n coins.

Two or more than two events are said to be mutually exclusive if the occurrence of one of the events excludes the occurrence of the otherThis can be better illustrated with the following examplesWhen a coin is tossed, we get either Head or Tail. Head and Tail cannot come simultaneously. Hence occurrence of Head and Tail are mutually exclusive events.When a die is rolled, we get 1 or 2 or 3 or 4 or 5 or 6. All these faces cannot come simultaneously. Hence occurrences of particular faces when rolling a die are mutually exclusive events.Note : If A and B are mutually exclusive events, A ∩ B = ϕ where ϕ represents empty set.Consider a die is thrown and A be the event of getting 2 or 4 or 6 and B be the event of getting 4 or 5 or 6. ThenA = {2, 4, 6} and B = {4, 5, 6}Here A ∩ B ≠ϕ. Hence A and B are not mutually exclusive events.