If A and B are independent and A ⊆ B then show that either P (A) = 0 or P (B) = 1.9. A single die is rolled; then n coins are tossed, where n is the number shown on the die.What is the probability of getting exactly two heads?10. The contents of Urn I, II and III are as follows:1 white, 2 red and 3 black balls,2 white, 2 red and 2 black balls and3 white, 1 red and 2 black balls.One urn is chosen at random and two balls are drawn. They happen to be white and red.What is the probability that they come from Urn I (or II or III)?

Sure, I can help with that. Here's how you can find the probability:

9. A single die is rolled; then n coins are tossed, where n is the number shown on the die. What is the probability of getting exactly two heads?

Step 1: Determine the total number of outcomes

When a die is rolled, each face has an equal probability of 1/6. The number of coins tossed depends on the outcome of the die roll.

Step 2: Determine the number of outcomes where exactly two heads are obtained

This can only happen when the die roll is 2, 3, 4, 5, or 6.

If the die roll is 2, then both coins must be heads, which has a probability of (1/2)^2 = 1/4.

If the die roll is 3, 4, 5, or 6, then we need to calculate the binomial probability of getting exactly two heads. The binomial probability formula is:

P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))

where n is the number of trials (the die roll), k is the number of successes (two heads), p is the probability of success on a single trial (1/2 for a coin toss), and C(n, k) is the number of combinations of n items taken k at a time.

For n = 3, 4, 5, or 6, we calculate this probability and sum them up.

Step 3: Calculate the total probability

The total probability is the sum of the probabilities for each possible die roll, each multiplied by the probability of that die roll (1/6).

10. The contents of Urn I, II and III are as follows: 1 white, 2 red and 3 black balls, 2 white, 2 red and 2 black balls and 3 white, 1 red and 2 black balls. One urn is chosen at random and two balls are drawn. They happen to be white and red. What is the probability that they come from Urn I (or II or III)?

Step 1: Determine the total number of outcomes

There are 3 urns, so the total number of outcomes is 3.

Step 2: Determine the number of outcomes where a white and a red ball are drawn

This can only happen if the chosen urn is Urn I, II, or III. We calculate the probability of drawing a white and a red ball from each urn, and sum these probabilities.

Step 3: Calculate the conditional probability

The conditional probability that the balls come from Urn I (or II or III) given that a white and a red ball were drawn is the probability of drawing a white and a red ball from Urn I (or II or III) divided by the total probability of drawing a white and a red ball.

This requires a detailed calculation for each urn and the final answer will depend on these calculations.