How do I find this probability The resistance value of a component is nominally 3300 Ω.
The standard deviation is 200 Ω.
Calculate the probability that the resistance value of a component is less than 3000 Ω.
Give your answer as a percentage to at least 2 significant figure (do not enter the percentage sign in your answer).
 A: The way to solve these types of problems is to use the normal distribution, if we can assume that our data are normally distributed. If we're asked for the probability of a range of values, then we need to calculate the probability of our value taking on any value in that range.
Here, we're asked for the probability of a component's resistance taking on any value less than 3,000 ohms, so we need to calculate the probability of it being 2,999 ohms, 2,998 ohms, et cetera. Let's call the resistance of the component X, so we need to calculate the probability that X is less than 3,000. To do this, we need the mean and variance of the underlying distribution, which we know is 3,300 ohms (for the mean) and 200 ohms (for the variance).
Once we know the value of our X, the mean of the normal distribution, and the variance of the normal distribution, we can calculate something called a z-statistic which is $z = \frac{X - \mu}{\sigma^2}$ where $\mu$ is the average and $\sigma^2$ is the variance. We can then take this value and look in a z-table (which almost every statistics textbook has) to figure out the probability of z being less than what we calculated. If you don't have a z-table, there are millions of them online that you can use.
A potential issue we might have is if our z-statistic is negative, since most z-tables only go from z=0 to z=4 (or some other number). In those situations, we will need to multiply our z-statistic by -1, which also flips the inequality. For example, if our z-statistic is -2 and we want $\mathrm{Pr}(z < -2)$, this is equivalent to $\mathrm{Pr}(z > 2)$. These will give the same probabilities because the normal distribution is symmetric. If your z-table doesn't give probabilities that z is greater than a value, then you can always just subtract the probability that z is less than that value from 1.
Like I mentioned above (and you know because you have the answer), the probability you should get is around 6.7%, which you get from looking up the probability of $z < -1.5$ or equivalently the probability that $z > 1.5$ or $1 - \mathrm{Pr}(z < 1.5)$. From my explanation above, you should be able to fill in all the details.
