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1. Two distinguishable fair dice, A and B, are rolled, and the values on the uppermost faces are noted.

a. What are the elements of the sample space?
b. What is the probability that the sum of the face values of the two dice is 10?
c. What is the probability that at least one 5 appears?

2. Let \( X \) be a continuous random variable with p.d.f. \( f_X(x) = 3x - 2 \) for \( 1 < x < a \).

a. Show that \( f(x) \) is a possible probability density function.
b. Find its cumulative distribution function.

5. In the standard gambler's ruin problem, with total stake \( a \) and gambler's stake \( k \), and the gambler's probability of winning each play is \( p \), calculate the probability of ruin in the following cases:

a. \( a = 100, k = 5, p = 0.6 \)
b. \( a = 80, k = 70, p = 0.45 \)
c. \( a = 50, k = 40, p = 0.5 \)

Also, find the expected duration in each case.

6. Find the probability that a random walk of 12 steps with probability \( p = 0.4 \) ends at:

a. position \( x = 8 \)
b. position \( x = -2 \)

7. Define the term Markov Chain.

8. Consider the transition diagram below. Let \( X_0 \sim (2/3, 0, 1/3, 0, 0, 0, 0) \).

What is the probability of the trajectory 1, 2, 3, 2, 1, 5?

Answer :

The sample space consists of all possible outcomes when rolling two distinguishable fair dice, with each outcome represented by an ordered pair of numbers. The probability of the sum of the face values being 10 can be calculated by counting the number of favorable outcomes and dividing it by the total number of outcomes. The probability of at least one 5 appearing can be calculated using the concept of complementary events.

1. The sample space for rolling two distinguishable fair dice consists of all possible outcomes, represented by ordered pairs of numbers ranging from 1 to 6. The probability of the sum of the face values being 10 can be calculated by determining the number of favorable outcomes (pairs that sum up to 10) and dividing it by the total number of outcomes (36). The probability of at least one 5 appearing can be calculated by finding the complement of the event "no 5 appears" and subtracting it from 1.

2. The given p.d.f. satisfies the non-negativity condition and integrates to 1 over its defined range, making it a valid probability density function. The cumulative distribution function can be obtained by integrating the p.d.f. from negative infinity to the given value of x.

5. In the standard gambler's ruin problem, the probability of ruin can be calculated by considering the probabilities of the gambler losing all their stake before reaching the target stake. The expected duration is found by summing the products of the number of steps and their corresponding probabilities for each possible outcome.

6. The probability that a random walk of 12 steps ends at a specific position can be calculated using the formula for the probability of reaching a specific position in a random walk. The formula takes into account the number of steps, the probability of moving in a specific direction, and the target position.

7. A Markov chain is a mathematical model that describes a sequence of events or states where the future state depends only on the current state and is independent of the past states. It involves a set of states, transition probabilities, and an initial probability distribution.

8. The probability of the trajectory 1,2,3,2,1,5 can be determined by multiplying the transition probabilities corresponding to each step in the trajectory. The initial probability distribution provides the starting point for calculating the probabilities.

Learn more about probability here:

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