Question

Consider the probability experiment of a sequence of independent trials (number of trials n=11), each trial is the rolling of a pair of 6-sided dice. suppose a random event A is the sum of the roll is an odd number and a random variable represents the number of successes that occur in the n trials. Find the probability of the success for a single trial, find the probability mass function Pn(m), m=1,n, find the mathematical expectation and variance of the random variable

Answer 1

Conditions

Consider the probability experiment of a sequence of independent trials (number of trials $n = 11$ ), each trial is the rolling of a pair of 6-sided dice. Suppose a random event A is the sum of the roll is an odd number and a random variable represents the number of successes that occur in the n trials. Find the probability of the success for a single trial, find the probability mass function $P_n(m)$ , $m = 1, n$ , find the mathematical expectation and variance of the random variable

Solution

Let's random variable $\xi$ is represents the number of successes that occur in the n trials.

Look at the table below:

Using Bernoulli's formula to find $p$ for each number of success.

$$P _ {n, m} = C _ {n} ^ {m} p ^ {m} q ^ {n - m}$$ $$P _ {1 1, 0} = C _ {1 2} ^ {0} p ^ {0} q ^ {1 1} = \frac {1 1 !}{1 0 ! 0 !} \left(\frac {1}{2}\right) ^ {1 1} = 0. 0 0 0 4 9$$ $$P _ {1 1, 1} = C _ {1 2} ^ {1} p ^ {1} q ^ {1 0} = 0. 0 0 5 3 7$$ $$P _ {1 1, 1 1} = 0. 0 0 0 4 9$$$$M (\xi) = \sum_ {i = 0} ^ {1 1} \xi_ {i} x _ {i} = 5. 5$$ $$D (\xi) = M \left(\xi^ {2}\right) - \left(M (\xi)\right) ^ {2}$$$$D (\xi) = 3 3 - 5. 5 ^ {2} = 2, 7 5$$