What is some application of variance in actuarial science/insurance risk?
I learn that a lot of applied math book of actuarial science have variance of probability distribution frequently.
Don't the books already have its probability for different problems?
Variance is one of the simplest measures of dispersion we have. Dispersion is important because it is a measure of how far off our random variable can be from the mean, and its calculation is nice and smooth (as opposed to, say, absolute deviation).
Given a measure of dispersion, you can start to think about risk loads. Since insurance contracts usually do not trade in liquid markets, insurance companies have to set a fixed price for them. This usually entails an estimate of expected losses, plus expenses, plus a "reasonable" provision for profit. A multiple of some measure of dispersion, like variance, can be a good starting point for a profit provision.
Note that current theory is moving away from dispersion and towards tail risk. This idea says that it is the extreme losses, "in the tail", which matter more for thinking about risk and profit. In liquid markets, dispersion makes sense because you can trade around the price, and so volatility (i.e. dispersion) is your main concern. But if you cannot reprice the risk, it makes sense to focus on the outcomes that do you the most harm, and that would be the tail risk.
EDIT: For example, if $X$ is a positive random variable representing an insurance loss, then:
$E(X)+\alpha Var(X)$
is a possible formula for a risk loaded pure premium (excluding expenses), where:
$E(X)$ is the expected loss of the contract
$Var(X)$ is the variance of the contract, and
$\alpha$ is a load factor to apply to the variance to change variance to profit load.
Note that there will be no consensus on what $\alpha$ should be, beyond what will sell in the market. Also, this model is disputable: some people will prefer using standard deviation (the square root of variance) because its unit are the same as the mean. But this gets us into a complex discussion on risk theory that is beyond the scope of your question.