Probability

Step 1: Find the total number of possible outcomes.
We need to form a four-digit number using the digits {1, 2, 3, 4, 5} without repetition.
The number of ways to choose and arrange 4 digits from 5 is given by the permutation formula .
Total outcomes = .

Step 2: Find the number of favorable outcomes.
A number is divisible by 3 if the sum of its digits is divisible by 3.
The sum of all five available digits is , which is divisible by 3.
To form a four-digit number, we must leave out one digit. For the sum of the remaining four digits to be divisible by 3, the digit we leave out must also be divisible by 3.
From the set {1, 2, 3, 4, 5}, the only digit divisible by 3 is 3. So, to form a number divisible by 3, we must use the digits {1, 2, 4, 5}.
The sum is , which is divisible by 3.
Alternatively, if we remove 1, sum is 14 (not div by 3). Remove 2, sum is 13 (not div by 3). Remove 4, sum is 11 (not div by 3). Remove 5, sum is 10 (not div by 3).
So, the only combination of four digits that works is {1, 2, 4, 5}. The number of four-digit numbers that can be formed using these four digits is .
Favorable outcomes = .

Step 3: Calculate the probability. Probability = .

Step 1: Define the events.
Let the events be as follows: - : The event that the transferred ball is red. - : The event that the drawn ball is black. We are required to find , the probability that the transferred ball is red, given that the drawn ball is black.

Step 2: Apply Bayes' Theorem.
Bayes' Theorem gives the following formula:

Step 3: Calculate the probabilities.
- : The probability that the transferred ball is red. There are 3 red balls out of a total of 10 balls in Bag I, so: - : The probability that a black ball is drawn from Bag II, given that a red ball was transferred from Bag I. After transferring a red ball, Bag II will have 3 red, 5 black, and 2 white balls, making a total of 10 balls. The probability of drawing a black ball is: - : The probability that a black ball is drawn from Bag II. This can occur in two ways: either a red ball or a black ball is transferred from Bag I. We calculate the probabilities of both cases: - If a red ball is transferred (event ), Bag II has 3 red, 5 black, and 2 white balls, making a total of 10 balls. The probability of drawing a black ball is . - If a black ball is transferred (event ), Bag II has 2 red, 6 black, and 2 white balls, making a total of 10 balls. The probability of drawing a black ball is . So, the total probability of drawing a black ball is: Simplifying:

Step 4: Calculate .
Now we can substitute the values into Bayes' Theorem:

Step 5: Conclusion.
Thus, the probability that the transferred ball is red is , and the correct answer is (b).

Step 1: Total number of balls.
The total number of balls in the bag is:

Step 2: Number of ways to choose 3 balls from 12.
The total number of ways to choose 3 balls from 12 is given by the combination formula:

Step 3: Number of ways to choose 3 balls excluding black balls.
If no black ball is chosen, then we must choose 3 balls from the 5 yellow and 3 green balls (8 balls in total). The number of ways to choose 3 balls from these 8 is:

Step 4: Calculate the probability.
The probability that no black ball is chosen is the ratio of favorable outcomes to total outcomes:

Step 5: Conclusion.
Thus, the probability that no black ball is chosen is , and the correct answer is (a).

Step 1: Use the law of total probability.
The total probability of getting heads is: Since the coin is chosen randomly, the probability of choosing each coin is . Thus: Thus, the correct answer is , corresponding to option (c).

Step 1: Total number of arrangements.
The total number of ways to arrange 5 captains is . Thus, the total arrangements are:

Step 2: Favorable arrangements.
Now, treat the captains of India and Australia as a block. This reduces the problem to arranging 4 blocks (the India-Australia block and the other 3 captains). The number of ways to arrange these 4 blocks is , and since India and Australia can be arranged within their block in ways, the total favorable arrangements are:

Step 3: Probability.
The probability that India and Australia stand next to each other is the ratio of favorable arrangements to total arrangements: Thus, the correct answer is , corresponding to option (b).