**Rule of Multiplication**

# Separate events

Events related to a phenomenon are separate if the events do not have any common elementary cases.

**Example 1.**

Are events *A = *“Take a card from a deck and it is a 6” and event *B *= “Take a card from a deck and it is a 7” separate?

Answer: Events *A *and* B* are separate because they do not have any common elementary cases.

Are events *A* = “Take a card from a deck and it is a 6 and event* D* = “Take a card from a deck and it is a spade” separate?

Answer: Events *A *and* D* are not separate events because they have one common elementary case, the 6 of spades.

**Example 2**. There are 655 students in a high school, 210 of who exercise, 135 who play music, and 355 who do not exercise or play music. What is the probability that a high school student will both play music and exercise?

355 + 210 + 135 - 655 = 45 students play music and exercise

The information can be presented in a so-called Venn diagram

Answer: A student plays music and exercise with a probability of 0,069

### Conditional probability

Conditional probability means the probability of event B on the condition that event A has already occurred. It is denoted P (B | A). Let P (A)> 0. The probability of event B on the condition that A has occurred is

**Example 3**.

This task is based on the data in Example 2.

Calculate the probability that a student who enjoys music will also exercise.

Answer: The probability that a student who plays music and also exercises is about 0,33.

### Rule of Multiplication for separate events

If we know that events A and B are independent, then

**Example 4**

Three cards are drawn from a deck and after each withdrawal the card is shuffled back into the deck.

a) Are the events separate?

b) What is the probability of getting three hearts in a row?

c) What is the probability of getting three aces in a row?

Answers

a) The events are separate because the card is always shuffled back into the deck, so previous draws of the card do not affect the probability of the following draws.

**Example 5.**

Three cards are drawn from a deck and the cards are not shuffled back into the deck after the drawings.

a) Are the events separate

b) What is the probability of getting three hearts in a row?

c) What is the probability of getting three aces in a row?

Answers

a) The events are not separate, because drawing previous cards from the deck reduces the number of cards in the deck and so affects the probability of the following draws.

Answer: The probability of getting three hearts in a row when the cards are not returned to the deck is 0,013.

Answer: The probability of getting three aces in a row when the cards are not returned to the deck is 0,00018.

## Complementary events

The complementary event of event *A* refers to all other elementary cases of some phenomenon other than event *A*. The sum of the probabilities of event A and its complementary event is one, so

**Example 6**

Write what is the complement of the event.

a) Event A = “roll a dice and get 6”

b) Event B = “roll a dice and not get more than 4”

c) Event C = “at least one student is late for class”

**Solution**

a) Counteraction = “get 5 or less”

b) Counteraction = “get 5 or more”

c) Countermeasure = “no student is late for class”

**Example 7**

Alma-Gunilla Heiskanen was on her way home from work. There are two bars on her way home, the 'Pub' and the 'Bistro'. Alma-Gunilla’s spouse Per-Ulf has noted her visits to the bar and ended up with the following separate probabilities. Alma-Gunilla has a 10% chance to visit the 'Pub' and a 20% chance to visit the 'Bistro'.

a) What is the probability that Alma-Gun will visit both bars on her way to home?

b) What is the probability that Alma-Gun will not visit either bar on her way home?

c) What is the probability that Alma-Gun will visit at least one bar on her way home?

It is known from the assignment that

P (“goes to the Pub”) = 0.10 P (“doesn't go to the Pub”) = 1 - 0.10 = 0.90

P (“goes to the Bistro”) = 0.20 P (“doesn’t go to the Bistro”) = 1 - 0.20 = 0.80

a) Visits both bars

Answer: Alma-Gunilla will visit both bars with a probability of 2%.

b) Does not visit either

Answer: Alma-Gunilla does not visit either bar with a probability of 72%.

c) The complement of the event “goes to at least one bar” is “does not go in either”, so

Answer: Alma-Gunilla will visit at least one bar on her way home with a probability of 0,28.

**Example 8**

2% of the products sold by a company have a defective colour.

a) What is the probability that none of the ten products randomly selected has a defective colour?

b) What is the probability that at least one of the ten randomly selected products has a defective colour.

a) The events are separate Since P (“defective”) = 0,02, then P (“not defective”) = 0,98 and

Answer: The probability of getting ten flawless products is 0,82

b) “At least one” questions are easiest to calculate using a complementary event.

Answer: The probability that at least one of the ten products has a colour defective is 0,18.

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