Understanding Probability and Variance

The question asks for the variance given the probability of winning a match is 0.7. This scenario describes a single event with two possible outcomes: winning or losing. In probability, such an event is known as a Bernoulli trial.

What is a Bernoulli Trial?

A Bernoulli trial is a random experiment with exactly two possible outcomes, typically labeled "success" and "failure," where the probability of success is the same every time the experiment is conducted. In this case:

• Success: Winning the match
• Failure: Losing the match

Probability of Success and Failure

We are given the probability of winning the match (success):

\(P(\text{Success}) = p = 0.7\)

The probability of losing the match (failure) is \(1 - p\). Let's call the probability of failure \(q\):

\(q = 1 - p\)

\(q = 1 - 0.7\)

\(q = 0.3\)

So, the probability of losing the match is 0.3.

Calculating the Variance of a Bernoulli Trial

For a single Bernoulli trial, the variance of the outcome (where success is assigned a value of 1 and failure a value of 0) is calculated using a simple formula:

\(\text{Variance} = p \times q\)

Where:

• \(p\) is the probability of success
• \(q\) is the probability of failure

Now, we can substitute the values of \(p\) and \(q\) that we have:

\(\text{Variance} = 0.7 \times 0.3\)

\(\text{Variance} = 0.21\)

Therefore, the variance for this probability of winning the match is 0.21.

Comparing with Options

Let's look at the given options:

1. 0.3
2. 0.21
3. 0.7
4. 0.31

Our calculated variance is 0.21, which matches option 2.

Conclusion on Variance Calculation

Based on the probability of winning the match (0.7), treated as a single Bernoulli trial, the variance is calculated as the product of the probability of success (0.7) and the probability of failure (0.3), resulting in a variance of 0.21.

Revision Table: Probability and Variance

Additional Information: Bernoulli vs. Binomial Distribution

It's helpful to understand the context of Bernoulli trials. While a single event is a Bernoulli trial, a sequence of independent Bernoulli trials forms a Binomial distribution. If the question had asked for the variance of the number of wins in, say, 10 matches, that would involve the Binomial distribution.

For a Binomial distribution \(B(n, p)\), where \(n\) is the number of trials and \(p\) is the probability of success in a single trial:

• Mean = \(n \times p\)
• Variance = \(n \times p \times q\)

However, since the question asks for "the value of variance" without specifying the number of matches (implicitly referring to the variance of the outcome of a single match or a single trial's result), we use the Bernoulli variance formula (\(p \times q\)), which is equivalent to the Binomial variance with \(n=1\).