Binomial Distribution

Step 1: Recall the formulas.
For a binomial distribution, the mean and variance are given by:

Step 2: Set up the system of equations.
We are given: From the first equation: Substitute this into the variance equation:

Step 3: Simplify and solve.

Step 4: Conclude.
Thus, , so the correct answer is option (ii).

Final Answer:

Step 1: Understand the distribution of .
Since three coins are tossed simultaneously, the number of heads, , can take values . The probability distribution of is binomial with trials (tosses) and probability of heads. The probability mass function for is: For , the probabilities are:

Step 2: Calculate the expected value .
The expected value of a binomial distribution is given by:

Step 3: Calculate the variance .
The variance of a binomial distribution is given by:

Final Answer: - Expected value - Variance