IN A THROW OF A DIE, A SUCCES IS DEFINED AS THE OUTCOME OF 1 OR 2. IF A DIE IS THROWN 12 TIMES
A, CACULATE THE PROBABILITY OF MORE THAN TWO SUCCESSES
B, AT MOST TWO SUCCESSES
C, OBTAIN THE PGF OF THE UNDERLYING RANDOM VARIABLE X AND USE IT TO DERIVE THE MEAN AND VARIANCE OF X
To calculate the probabilities, we need to first determine the sample space and the total number of possible outcomes.
The sample space in this case is the set of all possible outcomes when throwing a die. Since a die has six faces with numbers 1 to 6, the sample space can be represented as {1, 2, 3, 4, 5, 6}.
A) Probability of more than two successes:
To calculate the probability of getting more than two successes, we need to count the number of outcomes that are considered successes (1 or 2), and divide it by the total number of outcomes.
Number of successes (outcomes of 1 or 2) = 2
Number of trials = 12
The probability of success, p = Number of successes / Number of trials
So, the probability of more than two successes is:
P(X > 2) = 1 - P(X ≤ 2)
B) Probability of at most two successes:
To calculate the probability of getting at most two successes, we again count the number of outcomes that are considered successes (1 or 2), and divide it by the total number of outcomes.
Number of successes (outcomes of 1 or 2) = 2
Number of trials = 12
The probability of success, p = Number of successes / Number of trials
So, the probability of at most two successes is:
P(X ≤ 2) = 1 - P(X > 2)
C) Obtaining the PGF (Probability Generating Function) can help calculate the mean and variance of the underlying random variable X. The PGF is defined as the expected value of t raised to the power of X, where t is a parameter.
The PGF is calculated as:
G(t) = E(t^X) = Σ P(X = k) * t^k
Where P(X = k) is the probability mass function (PMF) of X, which can be calculated using the binomial distribution formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
In this case, since we only have two possible outcomes with equal probabilities, p = 1/3 for successes (1 or 2), and n = 12 (number of trials).
Using these values, we can substitute them into the PGF formula to calculate G(t). The mean and variance of X can then be derived from the PGF by taking the derivatives with respect to t.
Mean (μ) = G'(1)
Variance (σ^2) = G''(1) + G'(1) - (G'(1))^2
You can use the above formulas to calculate the exact values in parts A, B, and C for this specific scenario.