**Rule of Addition**

Previously, it was discussed that in probability, “AND” means multiplication. Now, let’s introduce some new information that “OR” means addition, as you may have already deduced from the title. The history of probability calculus and its beginnings can be seen from correspondence relating to the games of chance between the French mathematicians Pierre Fermat (1601 - 1655) and Blaise Pascal (1623 - 1662). For this historical reason, probability calculus tasks are often related to gambling. Below is a picture of a deck of cards

**Introduction example**

Calculate the probability that a card drawn from the deck is

a) a spade

b) a king

c) a spade and a king (i.e. the king of spades)

d) a spade or a king

a) There are a total of 13 spades from 52 cards in the deck, so

b) There are a total of 4 kings from a total of 52 cards, so

c) There is a total of 1 king of spades from a total of 52 cards, so

In this task first we added up the probabilities that the card is a spade, or that the card is a king. From this is deducted the alternative king of spades, as it would otherwise be counted twice. The king of spades belongs to both the spades and the kings.

Common addition rule

If events A and B have common elementary cases, then

Addition rule for independent events

If events A and B do not have any common elementary case, then

**Example 1**

Of the 30 students in 10th grade, 10 play music, 8 exercise, and 3 do both. Calculate the probability that a randomly selected student will play music or exercise?

The events “play music” and “exercise” are not independent of each other because they have common elementary cases of “play music and exercise”.

We will mark:

A = “play music”

B = “exercise”

Use the common addition rule as an aid and obtain the probability

Answer: A randomly selected student exercises or plays music with a probability of 0,50

**Example 2**

There are five green, six red and three blue balls in a box. What is the probability that when blindfolded, you lift a green or red ball out of the box?

There are 5 + 6 + 3 = 14 balls in total. We use the information that “OR” means a plus sign.

Answer: The probability that you get either a green or red ball is 11/14.

**Example 3**

You plant two seeds of which the chance of germination of seed A is 80% and that of seed B 60%. What is the probability that one of the seeds will germinate?

P (“A germinates”) = 0.80 P (“A does not germinate”) = 0.20

P (“B germinates”) = 0.60 P (“B does not germinate”) = 0.40

A probability calculation is generated using the information that “OR” means a plus sign and “AND” means a multiplier sign.

Answer: The probability that one of the seeds germinates is 0,44

**Example 4**

There are five green, six red and three blue balls in a box. What is the probability that when blindfolded, two balls of the same colour are lifted out of the box?

There are 5 + 6 + 3 = 14 balls in total. We use the information that “OR” means a plus sign and “AND” a multiplier. The calculation also takes into account the effect of the first draw on the probability of the second draw.

Answer: The probability of getting two balls of the same colour is 3/14.

**Example 5**

Per-Ulf Heiskanen has stated that during a commute, traffic lights will appear green with a 70% probability. The lights work independently. There are three sets of traffic lights along the commute. What is the probability that Per-Ulf will have to stop at traffic lights only once during his commute.

P (“green”) = 0,70

P (“red”) = 0,30

Of the traffic lights, red can be the first, second, or third of the lights on the way. So there are three options and the probability of each is the same.

Answer: The probability that Per-Ulf will have to stop once at the lights is 0,44

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