Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

Summation of Probabilities

In a lot of situations we are interested in the occurrence of events that consist of several data points, rather than just one sample point. The most common examples are when we look for the probability that an outcome is larger or smaller than a particular value. The probability of the compound event is then the sum of probabilities of the sample points that comprise the event. There are several simple rules for calculating compound events.  

Summation rule The probability of an event A is calculated by summing up the probabilities of the sample points in the sample space for A.

Example 1: If you toss a die, what are the chances of getting a number greater than 4?
Each of the outcomes has a probability of 1/6 and we have 2 numbers that are greater than four: 5 and 6, so we add these two probabilities to get the probability of observing a number larger than 4: 1/6+1/6 = 2 * 1/6 =1/3. So we have - as expected - a 33-percent chance of seeing a number greater than four.

Example 2: If we toss two coins, what is the probability of getting (1) exactly one head, and (2) at least one head? Our sample points are:
    value   probability
    KK     1/2*1/2 = 1/4
    KZ     1/2*1/2 = 1/4
    ZK     1/2*1/2 = 1/4
    ZZ     1/2*1/2 = 1/4
So the event of observing exactly one head consists of two sample points: HT and TH. We add their probabilities and get the probability of the event: 1/2. The event of observing at least one head consists of the sample points HT, TH, and HH. So the probability of the event is 3/4.